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Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems...

mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables...

mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of...

In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket...

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...

Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b...

mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties...

Look up algebra in Wiktionary, the free dictionary. Algebra may refer to: Elementary algebra Universal algebra Abstract algebra Linear algebra Relational...

mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure...

In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle...

geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is...

quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending...

measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the...

In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring R that is a finitely generated R-module. They are named after Emil Artin. Every...

In database theory, relational algebra is a theory that uses algebraic structures for modeling data and defining queries on it with well founded semantics...

mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional...

a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used...

In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center...

In set theory, the random algebra or random real algebra is the Boolean algebra of Borel sets of the unit interval modulo the ideal of measure zero sets...

In universal algebra, an abstract algebra A is called simple if and only if it has no nontrivial congruence relations, or equivalently, if every homomorphism...

In mathematics, an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection...

In commutative algebra, an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field...

mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive...

mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure...

Polyadic algebras (more recently called Halmos algebras) are algebraic structures introduced by Paul Halmos, designed to study first-order logic. Polyadic...

a category algebra is an associative algebra, defined for any locally finite category and commutative ring with unity. Category algebras generalize the...

Nonlinear algebra is the nonlinear analogue to linear algebra, generalizing notions of spaces and transformations coming from the linear setting. Algebraic geometry...

In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related...

algebra Enveloping algebra, of an associative algebra: see Associative algebra § Enveloping algebra Enveloping von Neumann algebra, of a C*-algebra This...

analysis, a uniform algebra A on a compact Hausdorff space X is a closed (with respect to the uniform norm) subalgebra of the C*-algebra C(X) (the continuous...

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative)...

In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted...

mathematics, the multiplier algebra, denoted by M(A), of a C*-algebra A is a unital C*-algebra that is the largest unital C*-algebra that contains A as an ideal...

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both...

special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weighted algebra). The study...

In mathematics, an octonion algebra or Cayley algebra over a field F is a unital composition algebra over F that has dimension 8 over F. In other words...

In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with...

these solutions. Pre-algebra Elementary algebra Boolean algebra Abstract algebra Linear algebra Universal algebra An algebraic equation is an equation...

In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of...

Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until...

especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced by...

In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢...

branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: Algebraic data type...

In mathematics, the concept of groupoid algebra generalizes the notion of group algebra. Given a groupoid ( G , ⋅ ) {\displaystyle (G,\cdot )} (in the...

Takiff algebra is a Lie algebra over a truncated polynomial ring. More precisely, a Takiff algebra of a Lie algebra g over a field k is a Lie algebra of the...

In abstract algebra, a Robbins algebra is an algebra containing a single binary operation ∨ {\displaystyle \lor } and a single unary operation ¬ {\displaystyle...

In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module...

algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures...

In algebraic logic, an action algebra is an algebraic structure which is both a residuated semilattice and a Kleene algebra. It adds the star or reflexive...

AW*-algebra is an algebraic generalization of a W*-algebra. They were introduced by Irving Kaplansky in 1951. As operator algebras, von Neumann algebras,...

In algebra, the Malvenuto–Poirier–Reutenauer Hopf algebra of permutations or MPR Hopf algebra is a Hopf algebra with a basis of all elements of all the...

In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann...

In algebra, Solomon's descent algebra of a Coxeter group is a subalgebra of the integral group ring of the Coxeter group, introduced by Solomon (1976)...

In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity:...

details are known about al-Khwarizmi's life. His popularizing treatise on algebra, compiled between 813 and 833 as Al-Jabr (The Compendious Book on Calculation...

polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may...

mathematics, vector algebra may mean: The operations of vector addition and scalar multiplication of a vector space The algebraic operations in vector...

In algebraic topology, a Steenrod algebra was defined by Henri Cartan (1955) to be the algebra of stable cohomology operations for mod p {\displaystyle...

In algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras...

abstract algebra, a structurable algebra is a certain kind of unital involutive non-associative algebra over a field. For example, all Jordan algebras are...

circle. Its Lie algebra is (more or less) the Witt algebra, whose central extension the Virasoro algebra (see Virasoro algebra from Witt algebra for a derivation...

mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex...

affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given...

also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector...

In mathematics, a quadratic algebra is an algebra over a ring for which the algebra extends the ring by a new element that satisfies a monic, quadratic...

filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and...

In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field...

the theory of algebras over a field, mutation is a construction of a new binary operation related to the multiplication of the algebra. In specific cases...

Quotient algebra may refer to: Specifically, quotient associative algebra in ring theory          or quotient Lie algebra Quotient (universal algebra) in the...

Difference algebra is a branch of mathematics concerned with the study of difference (or functional) equations from the algebraic point of view. Difference...

theory and functional analysis — the cylindrical σ-algebra or product σ-algebra is a type of σ-algebra which is often used when studying product measures...

up Boolean algebra in Wiktionary, the free dictionary. Boolean algebra is the algebra of truth values and operations on them. Boolean algebra may also refer...

algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group. The Hecke algebra can...

In abstract algebra, an algebra extension is the ring-theoretic equivalent of a group extension. Precisely, a ring extension of a ring R by an abelian...

In mathematics, a Nakayama algebra or generalized uniserial algebra is an algebra such that each left or right indecomposable projective module has a unique...

In mathematics, the group algebra can mean either A group ring of an abelian group over some commutative ring. A group algebra of a locally compact group...

path algebra is an algebra constructed from a directed graph. Leavitt path algebras generalize Leavitt algebras and may be considered as algebraic analogues...

planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor. They also provide an appropriate algebraic framework...

In mathematics: In abstract algebra and mathematical logic a derivative algebra is an algebraic structure that provides an abstraction of the derivative...

In operator algebras, the Toeplitz algebra is the C*-algebra generated by the unilateral shift on the Hilbert space l2(N). Identifying l2(N) with the Hardy...

In mathematics, the Iwasawa algebra Λ(G) of a profinite group G is a variation of the group ring of G with p-adic coefficients that take the topology...

In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital)...

theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov...

mathematics, the notion of cylindric algebra, developed by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This...

Complex algebra may refer to: A complex algebra (set theory), also known as field of sets Algebra over the complex numbers Algebra involving complex numbers...

theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice...

In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the...

In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any order) with multiplication being...

Effect algebras are partial algebras which abstract the (partial) algebraic properties of events that can be observed in quantum mechanics. Structures...

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins...

{b^{2}-4ac}}}{2a}}}}}} Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted...

specifically the field of algebra, Sklyanin algebras are a class of noncommutative algebra named after Evgeny Sklyanin. This class of algebras was first studied...

In abstract algebra, a Jordan algebra is a nonassociative algebra (with unit) over a field whose multiplication satisfies the following axioms: x y =...

Algebraic notation is the standard method of chess notation, used for recording and describing moves. It is based on a system of coordinates to uniquely...

abstract algebra, a partial algebra is a pair <A, P> where A is a set and P is a collection of partial operations on A. In universal algebra, when P consists...

In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups. It was first...

The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements...

branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context....

In abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. The multiplication...

In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general...

In mathematics, an Azumaya algebra is a generalization of central simple algebras to R {\displaystyle R} -algebras where R {\displaystyle R} need not...

In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N...

mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and...

In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. For a Lie algebra g {\displaystyle {\mathfrak...

mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld...

Hidden algebra provides a formal semantics for use in the field of software engineering, especially for concurrent distributed object systems. It supports...

In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra, and possibly the abelian aspects...

specifically in category theory, F-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates...

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also...

a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional...

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...

In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolic representation of projective...

− 1879) introduced his algebras as a tensor product ("compound of algebras") of quaternion algebras (and its even sub-algebra), a concept introduced by...

Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky (2002, 2003, 2007). A cluster algebra of rank n is an integral domain...

In functional analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of B(H), the ring of bounded linear operators on a separable...

The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial...

spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) of physics. Spacetime algebra provides...

operations on rational numbers do. Fields are fundamental algebraic structures that are widely used in algebra, number theory, and many other areas of mathematics...

In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory...

In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are...

In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by A(T), is the space of absolutely convergent Fourier series. Here...

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have x...

mathematics, the Cuntz algebra O n {\displaystyle {\mathcal {O}}_{n}} , named after Joachim Cuntz, is the universal C*-algebra generated by n {\displaystyle...

In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology...

several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding...

In the history of mathematics, the generality of algebra was a phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in...

mathematics, a topological algebra A {\displaystyle A} is an algebra and at the same time a topological space, where the algebraic and the topological structures...

In theoretical physics, a constraint algebra is a linear space of all constraints and all of their polynomial functions or functionals whose action on...

A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in...

A braid algebra can be: A Gerstenhaber algebra (also called an antibracket algebra). An algebra related to the braid group. This disambiguation page lists...

In mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that...

In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with...

Allen's interval algebra is a calculus for temporal reasoning that was introduced by James F. Allen in 1983. The calculus defines possible relations between...

In mathematics, a median algebra is a set with a ternary operation ⟨ x , y , z ⟩ {\displaystyle \langle x,y,z\rangle } satisfying a set of axioms which...

mathematics, particularly in the theory of C*-algebras, a uniformly hyperfinite, or UHF, algebra is a C*-algebra that can be written as the closure, in the...

In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size...

In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative...

for the study of algebra. Usually, Algebra I is taught in the 8th or 9th grade. As an intermediate stage after arithmetic, pre-algebra helps students pass...

enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal...

In mathematics, the Hecke algebra of a pair (G, K) of locally compact or reductive Lie groups is an algebra of measures under convolution. It can also...

In mathematics, a G-algebra can mean either An algebra over a field equipped with an algebraic representation π : G → A u t ( A ) {\displaystyle \pi :G\to...

semisimple algebra is an associative Artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson...

In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with...

supersymmetry algebra (or SUSY algebra) is a mathematical formalism for describing the relation between bosons and fermions. The supersymmetry algebra contains...

In abstract algebra, a rng (pronounced "rung" /rʌŋ/) or non-unital ring or pseudo-ring is an algebraic structure satisfying the same properties as a ring...

types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures...

A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative...

In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact...

Information Age. Shannon was among the first to describe the use of Boolean algebra—essential to all digital electronic circuits—and helped found the field...

In universal algebra, a basis is a structure inside of some (universal) algebras, which are called free algebras. It generates all algebra elements from...

especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra A {\displaystyle A} over the real or complex...

mathematics, the nil-Coxeter algebra, introduced by Fomin & Stanley (1994), is an algebra similar to the group algebra of a Coxeter group except that...

In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since...

education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level that is designed to prepare students for the...

Stone algebra if and only if its skeleton S(L) is a sublattice of L. Boolean algebras are Stone algebras, and Stone algebras are Ockham algebras. The open-set...

In mathematics and theoretical computer science, a Kleene algebra (/ˈkleɪni/ KLAY-nee; named after Stephen Cole Kleene) is a semiring that generalizes...

In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group Sn, studied by Nyman (2003). It consists of...

In algebra, the Pareigis Hopf algebra is the Hopf algebra over a field k whose left comodules are essentially the same as complexes over k, in the sense...

Algebra Felicia Blessett (born April 9, 1976), usually known as Algebra Blessett or just Algebra, is an American contemporary R&B singer. Blessett's mother...

mathematics, the Zhu algebra and the closely related C2-algebra, introduced by Yongchang Zhu in his PhD thesis, are two associative algebras canonically constructed...

including number theory (the study of integers and their properties), algebra (the study of operations and the structures they form), geometry (the study...

a Lie algebra g {\displaystyle {\mathfrak {g}}} is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie...

In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras...

an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions...

ring). *-algebra Affine Lie algebra Akivis algebra Algebra for a monad Albert algebra Alternative algebra AW*-algebra Azumaya algebra Banach algebra Birman–Wenzl...

In algebra, a Malcev-admissible algebra, introduced by Myung (1983), is a (possibly non-associative) algebra that becomes a Malcev algebra under the bracket...

In mathematics, a Jónsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set X onto the product X×X...

In abstract algebra, a Valya algebra (or Valentina algebra) is a nonassociative algebra M over a field F whose multiplicative binary operation g satisfies...

In mathematics, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties...

complement. Then we can define the Borel σ-algebra over X {\displaystyle X} to be the smallest σ-algebra containing all open sets of X {\displaystyle...

In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations...

Its subalgebras include diagram algebras such as the Brauer algebra, the Temperley–Lieb algebra, or the group algebra of the symmetric group. Representations...

derivation is a function on an algebra that generalizes certain features of the derivative operator. Specifically, given an algebra A {\displaystyle A} over...

Affine algebra may refer to: Affine Lie algebra, a type of Kac–Moody algebras The Lie algebra of the affine group Reduced finitely-generated algebra over...

Łukasiewicz–Moisil algebras (LMn algebras) were introduced in the 1940s by Grigore Moisil (initially under the name of Łukasiewicz algebras) in the hope of...

In mathematics, BCI and BCK algebras are algebraic structures in universal algebra, which were introduced by Y. Imai, K. Iséki and S. Tanaka in 1966, that...

In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...

Dirichlet algebra is a particular type of algebra associated to a compact Hausdorff space X. It is a closed subalgebra of C(X), the uniform algebra of bounded...

associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionals. In general, it is a non-commutative algebra. It...

and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics...

In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string...

In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation ⊕ {\displaystyle \oplus } , a unary...

split Lie algebra is a pair ( g , h ) {\displaystyle ({\mathfrak {g}},{\mathfrak {h}})} where g {\displaystyle {\mathfrak {g}}} is a Lie algebra and h <...

Moderne Algebra is a two-volume German textbook on graduate abstract algebra by Bartel Leendert van der Waerden (1930, 1931), originally based on lectures...

operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are...

mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero...

algebra is a Z-graded algebra for which xy = (−1)deg(x)deg(y)yx for all nonzero homogeneous elements x and y (i.e. it is an anticommutative algebra)...

Roman 2008, p. 115, §4 Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1 Irving Kaplansky...

In algebra, a Taft Hopf algebra is a Hopf algebra introduced by Earl Taft (1971) that is neither commutative nor cocommutative and has an antipode of large...

In algebraic topology, through an algebraic operation (dualization), there is an associated commutative algebra from the noncommutative Steenrod algebras...

a quotient algebra is the result of partitioning the elements of an algebraic structure using a congruence relation. Quotient algebras are also called...

In mathematical physics, the Dirac algebra is the Clifford algebra Cl 1 , 3 ( C ) {\displaystyle {\text{Cl}}_{1,3}(\mathbb {C} )} . This was introduced...

Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below). The center of SU(n)...

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants...

In mathematics, particularly in the study of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of...

algebra is an extension of vector algebra, providing additional algebraic structures on vector spaces, with geometric interpretations. Vector algebra...

In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal...

In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the finite-dimensional Clifford algebras for a nondegenerate...

\mathbb {C} } is 0. A Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } -algebra is equivalently a ring whose characteristic divides n. This is because...

place-value system to include decimal fractions, the systematised study of algebra and advances in geometry and trigonometry. The medieval Islamic world underwent...

In algebra and logic, a modal algebra is a structure ⟨ A , ∧ , ∨ , − , 0 , 1 , ◻ ⟩ {\displaystyle \langle A,\land ,\lor ,-,0,1,\Box \rangle } such that...

algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is...

In mathematics, an exp algebra is a Hopf algebra Exp(G) constructed from an abelian group G, and is the universal ring R such that there is an exponential...

algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are...

Cantor algebra, named after Georg Cantor, is one of two closely related Boolean algebras, one countable and one complete. The countable Cantor algebra is...

In mathematics, a universal geometric algebra is a type of geometric algebra generated by real vector spaces endowed with an indefinite quadratic form...

In mathematics, a weak Lie algebra bundle ξ = ( ξ , p , X , θ ) {\displaystyle \xi =(\xi ,p,X,\theta )\,} is a vector bundle ξ {\displaystyle \xi \,}...

problems that are solved through algebraic equations. Joseph-Louis Lagrange called Diophantus "the inventor of algebra"; his exposition became the standard...

Map algebra is an algebra for manipulating geographic data, primarily fields. Developed by Dr. Dana Tomlin and others in the late 1970s, it is a set of...

analysis, a Banach function algebra on a compact Hausdorff space X is unital subalgebra, A, of the commutative C*-algebra C(X) of all continuous, complex-valued...

representing commutative Banach algebras as algebras of continuous functions; the fact that for commutative C*-algebras, this representation is an isometric...

of mathematics known as universal algebra (and in its applications), a subdirectly irreducible algebra is an algebra that cannot be factored as a subdirect...

representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms...

Loomis and Shlomo Sternberg. The official title of the course is Studies in Algebra and Group Theory (Math 55a) and Studies in Real and Complex Analysis (Math...

In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory T consists of the equivalence classes of sentences of...

apart from the elementary symbolic algebra: Expectation algebra, Variance algebra, Covariance algebra, Moment algebra, etc. Considering two random variables...

In mathematics, a Borchers algebra, Borchers–Uhlmann algebra, or BU-algebra is the tensor algebra of a vector space, often a space of smooth test functions...

graph C*-algebra is a universal C*-algebra constructed from a directed graph. Graph C*-algebras are direct generalizations of the Cuntz algebras and Cuntz-Krieger...

a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct...

In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field and plays a key role in the theory of central simple...

sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...

"two-by-three matrix", a 2 × 3 matrix, or a matrix of dimension 2 × 3. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric...

algebra Lie superalgebra Abelian Lie algebra Affine Lie algebra Anyonic Lie algebra Compact Lie algebra Complex Lie algebra Exceptional Lie algebra Finite-dimensional...

with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being...

mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers), variables, and the basic algebraic operations:...

In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism...

Elements of Algebra is an elementary mathematics textbook written by mathematician Leonhard Euler around 1765 in German. It was first published in Russian...

noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix...

In mathematical physics, a super Virasoro algebra is an extension of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There...

mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L ∞ {\displaystyle L_{\infty }} -algebra) is a generalisation of the concept...

In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. For example, in a signature...

This glossary of linear algebra is a list of definitions and terms relevant to the field of linear algebra, the branch of mathematics concerned with linear...

Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument...

In mathematics, a restricted Lie algebra (or p-Lie algebra) is a Lie algebra over a field of characteristic p>0 together with an additional "pth power"...

groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions...

in probability theory and ergodic theory, the invariant sigma-algebra is a sigma-algebra formed by sets which are invariant under a group action or dynamical...

algebra is an algebraic structure introduced by Beilinson and Drinfeld (2004) as a rigorous version of the rather vague concept of a chiral algebra in...

In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework...

homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often...

Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which...

In mathematics, an isotopy from a possibly non-associative algebra A to another is a triple of bijective linear maps (a, b, c) such that if xy = z then...

mathematics, a universal C*-algebra is a C*-algebra described in terms of generators and relations. In contrast to rings or algebras, where one can consider...

89–90) introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra H4 is a certain 4-dimensional quotient of it that is neither...

In commutative algebra, the Rees algebra or Rees ring of an ideal I in a commutative ring R is defined to be R [ I t ] = ⨁ n = 0 ∞ I n t n ⊆ R [ t ] ...

Informally in mathematical logic, an algebraic theory is a theory that uses axioms stated entirely in terms of equations between terms with free variables...

Algebra is a graduate-level textbook on abstract algebra written by Serge Lang and was originally published by Addison-Wesley in 1965. Its intended audience...

In mathematics, quadratic Jordan algebras are a generalization of Jordan algebras introduced by Kevin McCrimmon (1966). The fundamental identities of the...

the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that...

In abstract algebra, a cellular algebra is a finite-dimensional associative algebra A with a distinguished cellular basis which is particularly well-adapted...

of nonassociative algebras are flexible: Alternative algebras Lie algebras Jordan algebras (which are commutative) Okubo algebras In the world of magmas...

In mathematics, the Hecke algebra is the algebra generated by Hecke operators, which are named after Erich Hecke. The algebra is a commutative ring. In...

In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and...

areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A that is simple, and for which the center...

mathematics, a basic algebraic operation is a mathematical operation similar to any one of the common operations of elementary algebra, which include addition...

In algebra, let g be a Lie algebra over a field K. Let further ξ ∈ g ∗ {\displaystyle \xi \in {\mathfrak {g}}^{*}} be a one-form on g. The stabilizer...

In mathematics, an algebra such as ( R , + , ⋅ ) {\displaystyle (\mathbb {R} ,+,\cdot )} has multiplication ⋅ {\displaystyle \cdot } whose associativity...

In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism A → k {\displaystyle A\to k} , typically...

The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky...

Algebra i Logika (English: Algebra and Logic) is a peer-reviewed Russian mathematical journal founded in 1962 by Anatoly Ivanovich Malcev, published by...

In mathematics, a Malcev Lie algebra, or Mal'tsev Lie algebra, is a generalization of a rational nilpotent Lie algebra, and Malcev groups are similar...

In algebra, an operad algebra is an "algebra" over an operad. It is a generalization of an associative algebra over a commutative ring R, with an operad...

geometric algebra is a specific algebraic structure. The term is also used as a blanket term for the theory of geometric algebras. Geometric algebra may also...

studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements...

higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra. A first...

In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture...

In abstract algebra, an associative algebra A {\displaystyle A} over a ring R {\displaystyle R} is called finite if it is finitely generated as an R {\displaystyle...

number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the...

Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borcherds–Kac–Moody algebras, BKM algebras, or Borcherds algebras. The best known example...

In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from...

modules, Lie algebras, and other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences...

Topological Boolean algebra may refer to: In abstract algebra and mathematical logic, topological Boolean algebra is one of the many names that have been...

Abstrakt Algebra was a Swedish experimental metal band with influences from power metal and doom metal. It was founded by bassist Leif Edling in 1994...

In abstract algebra, a Koszul algebra R {\displaystyle R} is a graded k {\displaystyle k} -algebra over which the ground field k {\displaystyle k} has...

Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as parameters...

In mathematics, a normed algebra A is an algebra over a field which has a sub-multiplicative norm: ∀ x , y ∈ A ‖ x y ‖ ≤ ‖ x ‖ ‖ y ‖ . {\displaystyle...

In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector...

nuclear C*-algebra is a C*-algebra A such that for every C*-algebra B the injective and projective C*-cross norms coincides on the algebraic tensor product...

Algebra tiles, also known as Algetiles, or Variable Blocks, are mathematical manipulatives that allow students to better understand ways of algebraic...

mathematics, the term Beurling algebra is used for various different algebras introduced by Arne Beurling (1949). Usually it is an algebra of periodic functions...

In the history of mathematics, Egyptian algebra, as that term is used in this article, refers to algebra as it was developed and used in ancient Egypt...

Fourier and related algebras occur naturally in the harmonic analysis of locally compact groups. They play an important role in the duality theories of...

theories define an infinite-dimensional Lie algebra called a current algebra. Mathematically these are Lie algebras consisting of smooth maps from a manifold...

In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions...

Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such...

glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary...

algebra, and hence of the Lie group Sp ⁡ ( 2 n , F ) {\displaystyle \operatorname {Sp} (2n,\mathbb {F} )} , is n {\displaystyle n} . The Lie algebra of...

spin group or of the associated Clifford algebra. After choosing a matrix realization of the Clifford algebra, spinors may be represented concretely as...

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as...

Poincaré algebra is replaced by a supersymmetry algebra which is a Z2-graded Lie algebra extending the Poincaré algebra. The structure of such an algebra is...

German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental...

In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0 {\displaystyle P=0} , where P is a polynomial, usually with...

In the area of mathematics known as universal algebra, a clone is a set C of finitary operations on a set A such that C contains all the projections πkn:...

The term "information algebra" refers to mathematical techniques of information processing. Classical information theory goes back to Claude Shannon....

the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field...

In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by Dorothy Maharam...

In linear algebra, the trace of a square matrix A, denoted tr(A), is defined as a sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle...

known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky...

{Spin} (1,3)} . The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More...

two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition...

In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure A = (A, ∨, ∧, 0, 1, ¬) such...

In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are 3 x...

In mathematics, a Ringel–Hall algebra is a generalization of the Hall algebra, studied by Claus Michael Ringel (1990). It has a basis of equivalence classes...

composition algebras frequently applied in mathematical physics. The Cayley–Dickson construction defines a new algebra as a Cartesian product of an algebra with...

Algebraic statistics is a branch of mathematical statistics that focuses on the use of algebraic, geometric, and combinatorial methods in statistics. While...

In mathematics, quantum algebra is the study of noncommutative analogues and generalizations of commutative algebras, especially those arising in Lie theory...

Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0. Lay, David C. (2005), Linear Algebra and Its Applications (3rd ed...

Algebra and Tiling: Homomorphisms in the Service of Geometry is a mathematics textbook on the use of group theory to answer questions about tessellations...

Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which...

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more...

Álgebra, commonly known as Álgebra de Baldor (Spanish: Baldor's Algebra), is a book by the Cuban mathematician, lawyer, and professor Aurelio Baldor....

In mathematics, weak bi-algebras are a generalization of bialgebras that are both algebras and coalgebras but for which the compatibility conditions between...

possibilities. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras. The theory of composition algebras has subsequently...

In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R {\displaystyle R} which satisfies the Rota–Baxter...

Flag algebras are an important computational tool in the field of graph theory which have a wide range of applications in homomorphism density and related...

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various...

correctness of the counterexample is easy to verify with any computer algebra system. It has not been revealed, however, how it was found. Nevertheless...

algebra in Wiktionary, the free dictionary. In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures...

Almucabola), commonly abbreviated Al-Jabr or Algebra (Arabic: الجبر), is an Arabic-language mathematical treatise on algebra written in Baghdad around 820 by Al-Khwarizmi...

Birman–Murakami–Wenzl (BMW) algebra, introduced by Joan Birman and Hans Wenzl (1989) and Jun Murakami (1987), is a two-parameter family of algebras C n ( ℓ , m ) {\displaystyle...

In mathematics, a projectionless C*-algebra is a C*-algebra with no nontrivial projections. For a unital C*-algebra, the projections 0 and 1 are trivial...

mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential...

mathematics, a Hecke algebra is classically the algebra of Hecke operators studied by Erich Hecke. It may also refer to one of several algebras generalizing the...

unique. The term isomorphism is mainly used for algebraic structures and categories. In the case of algebraic structures, mappings are called homomorphisms...

and some contemporary developments are closely related to Mikio Sato's algebraic analysis. In the mathematics of the nineteenth century, aspects of generalized...

element theory, the calculus of variations, wavelet analysis and linear algebra. He has made many contributions to mathematics education, including publishing...

specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions ƒ : D → C {\displaystyle...

central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. In a vector space, the...

In mathematics, a Lie algebra has been generalized in several ways. A graded Lie algebra is a Lie algebra with grading. When the grading is Z / 2 {\displaystyle...

mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive if...

mathematics, a modular Lie algebra is a Lie algebra over a field of positive characteristic. The theory of modular Lie algebras is significantly different...

called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only...

theory, the concept of weights of an algebra A over a field F is a generalisation of that of eigenvalues. They are algebra homomorphisms from A to F, or equivalently...

In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a...

In algebra, a noncommutative Jordan algebra is an algebra, usually over a field of characteristic not 2, such that the four operations of left and right...

In universal algebra, a congruence-permutable algebra is an algebra whose congruences commute under composition. This symmetry has several equivalent characterizations...

constants forms a Lie algebra under the Poisson bracket. This Lie algebra is a one-dimensional central extension of the commutative Lie algebra R 2 n {\displaystyle...

In mathematics, Banach algebra cohomology of a Banach algebra with coefficients in a bimodule is a cohomology theory defined in a similar way to Hochschild...

In differential geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the...

operator algebras, the enveloping von Neumann algebra of a C*-algebra is a von Neumann algebra that, in some sense, contains all the operator-algebraic information...

Traditionally, algebra instruction has been postponed until adolescence. However, data of early algebra researchers shows ways to teach algebraic thinking much...

monad is a monoid in a certain category, A monad as a tool for studying algebraic gadgets; for example, a group can be described by a certain monad. Monads...

Hilbert algebras and left Hilbert algebras occur in the theory of von Neumann algebras in: Commutation theorem for traces § Hilbert algebras Tomita–Takesaki...

numbers. In algebraic number theory, integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In...

group Hopf algebra of a given group is a certain construct related to the symmetries of group actions. Deformations of group Hopf algebras are foundational...

the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is G L ( n , R ) {\displaystyle...

In algebra, the Nichols algebra of a braided vector space (with the braiding often induced by a finite group) is a braided Hopf algebra which is denoted...

operator algebra A is an operator algebra that has enough invariant subspaces to characterize it. Formally, A is reflexive if it is equal to the algebra of...

In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton...

rectangular bands and pairs of nonempty sets. Rectangular bands are also the T-algebras, where T is the monad on Set with T(X)=X×X, T(f)=f×f, η X {\displaystyle...

In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by Erik Verlinde (1988). It is defined to have basis of elements...

Strip Algebra is a set of elements and operators for the description of carbon nanotube structures, considered as a subgroup of polyhedra, and more precisely...

In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal...

Euclidean space, an algebraic variety is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields...

Algebra Smart is a software CD-ROM from The Princeton Review. It is for ages 14 and up. Algebra Smart consists of 12 lessons which use video clips of...

to finding a (quantum) algebra whose classical limit is a given (classical) algebra such as a Lie algebra or a Poisson algebra. Intuitively, a deformation...

In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate...

In mathematics, a Hodge algebra or algebra with straightening law is a commutative algebra that is a free module over some ring R, together with a given...

supercommutative (associative) algebra (sometimes termed a commutative superalgebra) is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous...

mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures...

In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that x...

of computer algebra systems (CAS). A CAS is a package comprising a set of algorithms for performing symbolic manipulations on algebraic objects, a language...

abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and...

respect to the σ-algebra generated by all left-continuous adapted processes. This σ-algebra is also called the predictable σ-algebra. Every deterministic...

In mathematics, BF-algebras are a class of algebraic structures arising out of a symmetric "yin yang" concept for Bipolar Fuzzy logic, the name was introduced...

In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of...

In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used...

In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix generated from A by removing one or more of its rows and columns...

algebra (also called an algebra of finite type) over a (commutative) ring R {\displaystyle R} , or a finitely generated R {\displaystyle R} -algebra for...

The Algebra Project is a national U.S. mathematics literacy program aimed at helping low-income students and students of color achieve the mathematical...

rational numbers, real numbers, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 results in 0, and consequently...

In algebraic geometry, given a linear algebraic group G over a field k, a distribution on it is a linear functional k [ G ] → k {\displaystyle k[G]\to...

Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric...

(isomorphism classes of) certain algebraic objects called Sullivan minimal models, which are commutative differential graded algebras over the rational numbers...

a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure...

In algebra, a commutative k-algebra A is said to be 0-smooth if it satisfies the following lifting property: given a k-algebra C, an ideal N of C whose...

postulate. Abstract algebra The part of algebra devoted to the study of algebraic structures in themselves. Occasionally named modern algebra in course titles...

This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations...

In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive...

In mathematics, an O*-algebra is an algebra of possibly unbounded operators defined on a dense subspace of a Hilbert space. The original examples were...

the real Lie algebra s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} , which exponentiates to the special unitary group SU(2). The algebra generated by...

Examples include associative algebras, Lie algebras, and Okubo algebras. Malcev-admissible algebra Jordan-admissible algebra Okubo 1995, p. 19 Albert, A...

In abstract algebra, a subset S {\displaystyle S} of a field L {\displaystyle L} is algebraically independent over a subfield K {\displaystyle K} if the...

a timeline of key developments of algebra: Mathematics portal History of algebra – Historical development of algebra Archibald, Raymond Clare (December...

Algebra: Chapter 0 is a graduate abstract algebra textbook written by Paolo Aluffi. The book was first published in 2009 by the American Mathematical...

In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together...

In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if its lower central series terminates in the zero subalgebra. The lower...

describe an algebraic structure which in some sense cannot be divided by a smaller structure of the same type. Put another way, an algebraic structure is...

In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove...

subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems...

Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework...

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations...

affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were...

analysis, a branch of mathematics, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute...

categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting...

In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of...

This is a list of algebraic topology topics. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces...

theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber...

The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types A n {\displaystyle A_{n}} , B n {\displaystyle...

enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra g {\displaystyle {\mathfrak {g}}} , the quantum enveloping algebra is...

In Boolean algebra, the inclusion relation a ≤ b {\displaystyle a\leq b} is defined as a b ′ = 0 {\displaystyle ab'=0} and is the Boolean analogue to...

In probability theory, a conditional event algebra (CEA) is an alternative to a standard, Boolean algebra of possible events (a set of possible events...

algebra is an algebraic structure first introduced by Beilinson and Drinfel'd in an algebro-geometric setting as a reformulation of chiral algebras and...

algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups)...

order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different...

A quadratic Lie algebra is a Lie algebra together with a compatible symmetric bilinear form. Compatibility means that it is invariant under the adjoint...

abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over...

In mathematics, an ultragraph C*-algebra is a universal C*-algebra generated by partial isometries on a collection of Hilbert spaces constructed from...

include Boolean arithmetic; The primary algebra (Chapter 6 of LoF), whose models include the two-element Boolean algebra (hereinafter abbreviated 2), Boolean...

In homological algebra, a branch of mathematics, a matrix factorization is a tool used to study infinitely long resolutions, generally over commutative...

mathematics, if A is an associative algebra over K, then an element a of A is an algebraic element over K, or just algebraic over K, if there exists some non-zero...

group (or supergroup), and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra (or Lie superalgebra). By extension, the...

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space...

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite...

Journal of Algebra (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier...

treated in the series include set theory, abstract algebra, topology, analysis, Lie groups and Lie algebras. The unusual singular "mathématique" (mathematic)...

group. The respective Lie algebra is called the Poincaré algebra. It is possible to extend this algebra to a supersymmetry algebra, which is a Z 2 {\displaystyle...

In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly...

Communications in Algebra is a monthly peer-reviewed scientific journal covering algebra, including commutative algebra, ring theory, module theory, non-associative...

abstract algebra, a quasi-free algebra is an associative algebra that satisfies the lifting property similar to that of a formally smooth algebra in commutative...

2s(F). The Stufe of a quadratically closed field is 1. The Stufe of an algebraic number field is ∞ {\displaystyle \infty } , 1, 2 or 4 (Siegel's theorem)...

In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R i...

In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott...

In database theory, Imieliński–Lipski algebra is an extension of relational algebra onto tables with different types of null values. It is used to operate...

In algebra, the Kantor–Koecher–Tits construction is a method of constructing a Lie algebra from a Jordan algebra, introduced by Jacques Tits (1962), Kantor (1964)...

In algebra, given a commutative ring R, the graded-symmetric algebra of a graded R-module M is the quotient of the tensor algebra of M by the ideal I generated...

mathematics, a generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann...

a consequence, a product of two negative numbers is positive. For an algebraic proof of this result, start with the equation 0 = −1 ⋅ 0 = −1 ⋅ [1 + (−1)]...

In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several...

In abstract algebra, a commutant-associative algebra is a nonassociative algebra over a field whose multiplication satisfies the following axiom: ( [ A...

mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus...

operator-algebraic methods based on C*-algebras, von Neumann algebras, and spectral triples; algebraic approaches to noncommutative rings and graded algebras;...

In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure. It follows that...

notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of data structures...

Linear Algebra is a 1966 mathematics textbook by Serge Lang. The third edition of 1987 covers fundamental concepts of vector spaces, matrices, linear...

In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for...

Plane-based geometric algebra is an application of Clifford algebra to modelling planes, lines, points, and rigid transformations. Generally this is with...

subject. For the items in commutative algebra (the theory of commutative rings), see Glossary of commutative algebra. For ring-theoretic concepts in the...

needed] The term algebraic coding theory denotes the sub-field of coding theory where the properties of codes are expressed in algebraic terms and then...

earlier work on algebra, tracing the development of "double" algebra, essentially geometric algebra, from arithmetic through symbolical algebra, illustrated...

In mathematics, specifically in functional analysis, a Banach algebra, A, is amenable if all bounded derivations from A into dual Banach A-bimodules are...

The Algebra of Ice is a BBC Books original novel written by Lloyd Rose and based on the long-running British science fiction television series Doctor...

In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together: n4 = n × n × n × n. Fourth powers...

finite-dimensional (AF) C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality...

for manipulations of tensors arise as an extension of linear algebra to multilinear algebra. In differential geometry, an intrinsic[definition needed] geometric...

A Lie conformal algebra is in some sense a generalization of a Lie algebra in that it too is a "Lie algebra," though in a different pseudo-tensor category...

group, Spin(n) therefore shares its dimension, ⁠n(n − 1)/2⁠, and its Lie algebra with the special orthogonal group. For n > 2, Spin(n) is simply connected...

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both...

subalgebra of the universal enveloping algebra of a semisimple Lie algebra to the universal enveloping algebra of a subalgebra. A particularly important...

Similarly the term "algebra over X {\displaystyle X} " is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings...

finite-dimensional hereditary algebra over K. Conversely, if K is algebraically closed, then any finite-dimensional, hereditary, associative algebra over K is Morita...

In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector...

In mathematics, specifically commutative algebra, a divided power structure is a way of introducing items with similar properties as expressions of the...

Nichols algebra is a Hopf algebra in a braided category assigned to an object V in this category (e.g. a braided vector space). The Nichols algebra is a...

abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain...

quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Ukrainian mathematician Vladimir Drinfeld in 1989. A quasi-Hopf algebra is...

In the study of geometric algebras, a k-blade or a simple k-vector is a generalization of the concept of scalars and vectors to include simple bivectors...

Metaplectic groups also occur in quantum mechanics. Algebra extension Lie algebra extension Virasoro algebra HNN extension Group contraction Extension of a...

an algebraic lattice. Also, a kind of converse holds: Every algebraic lattice is isomorphic to Sub(A) for some algebra A. There is another algebraic lattice...

cohomology of an algebra may refer to Banach algebra cohomology of a bimodule over a Banach algebra Cyclic homology of an associative algebra Group cohomology...

group on Minkowski space uses biquaternions, which form a composition algebra. The isometry property of Lorentz transformations holds according to the...

physics, the Duffin–Kemmer–Petiau (DKP) algebra, introduced by R.J. Duffin, Nicholas Kemmer and G. Petiau, is the algebra which is generated by the Duffin–Kemmer–Petiau...

tensor algebra can be constructed as quotients: these include the exterior algebra, the symmetric algebra, the Clifford algebra, the Weyl algebra, and the...

field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0: g ≃ g 0...

In homological algebra, a monad is a 3-term complex A → B → C of objects in some abelian category whose middle term B is projective, whose first map A → B...

abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra H...

In linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by...

The σ-algebra of τ-past, (also named stopped σ-algebra, stopped σ-field, or σ-field of τ-past) is a σ-algebra associated with a stopping time in the theory...

In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear...

In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular...

Geometric Algebra is a book written by Emil Artin and published by Interscience Publishers, New York, in 1957. It was republished in 1988 in the Wiley...

any set Ω {\displaystyle \Omega \,} (also called sample space) and a σ-algebra F {\displaystyle {\mathcal {F}}\,} on it, a measure P {\displaystyle \mathbb...

an algebraic structure. Formally, a filtration is an indexed family ( S i ) i ∈ I {\displaystyle (S_{i})_{i\in I}} of subobjects of a given algebraic structure...

In mathematics, an algebraic representation of a group G on a k-algebra A is a linear representation π : G → G L ( A ) {\displaystyle \pi :G\to GL(A)}...

In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature ⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩, where ⟨A, ·, +, ',...

In algebra, Freudenthal algebras are certain Jordan algebras constructed from composition algebras. Suppose that C is a composition algebra over a field...

being the use of the corresponding 'infinitesimal' representations of Lie algebras. A complex representation of a group is an action by a group on a finite-dimensional...

In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of H ⊗ H {\displaystyle H\otimes H} such that R   Δ (...

In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a...

general linear Lie algebra all of whose elements are semisimple (or diagonalizable over an algebraically closed field). Over an algebraically closed field,...

In algebra, a subfield of an algebra A over a field F is an F-subalgebra that is also a field. A maximal subfield is a subfield that is not contained...

ordinary differential equations by the Runge–Kutta method. It arose from an algebraic formalism involving rooted trees that provides formal power series solutions...

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of...

In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical n i l ( g ) {\displaystyle {\mathfrak {nil}}({\mathfrak...

The Algebra of Infinite Justice (2001) is a collection of essays written by Booker Prize winner Arundhati Roy. The book discusses a wide range of issues...

The International Journal of Algebra and Computation is published by World Scientific, and contains articles on general mathematics, as well as: Combinatorial...

symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures...

In mathematics, Zhegalkin algebra is a set of Boolean functions defined by the nullary operation taking the value 1 {\displaystyle 1} , use of the binary...

Algebra & Number Theory is a peer-reviewed mathematics journal published by the nonprofit organization Mathematical Sciences Publishers. It was launched...

type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other...

In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties...

In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)...

In abstract algebra, the total algebra of a monoid is a generalization of the monoid ring that allows for infinite sums of elements of a ring. Suppose...

In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations...

The Casio Algebra FX series was a line of graphing calculators manufactured by Japanese electronics company Casio Computer Co., Ltd from 1999 to 2003...

name "algebra of physical space" (APS) originally stems from the use of the biquaternions via its definition as the real Clifford or geometric algebra Cl3...

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that "division" is always possible...

field F is algebraically closed if every non-constant polynomial with coefficients in F has a root in F. In other words, a field is algebraically closed if...

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free...

In algebraic geometry, a sheaf of algebras on a ringed space X is a sheaf of commutative rings on X that is also a sheaf of O X {\displaystyle {\mathcal...

and songwriter, best known for her work with the new wave musical group Algebra Suicide, along with her husband Don Hedeker. Lydia Tomkiw was born in Chicago's...