# Category: Algebras

Filtered algebra
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory.
CCR and CAR algebras
In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermi
Duffin–Kemmer–Petiau algebra
In mathematical physics, the Duffin–Kemmer–Petiau algebra (DKP algebra), introduced by R.J. Duffin, Nicholas Kemmer and G. Petiau, is the algebra which is generated by the Duffin–Kemmer–Petiau matrice
In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important r
R-algebroid
In mathematics, R-algebroids are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie al
Semisimple algebra
In ring theory, a branch of mathematics, a 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 Jacobso
Beurling algebra
In mathematics, the term Beurling algebra is used for different algebras introduced by Arne Beurling, usually it is an algebra of periodic functions with Fourier series ExampleWe may consider the alge
Supercommutative algebra
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements x, y we have where |x| denotes the grade of the element
Double affine Hecke algebra
In mathematics, a double affine Hecke algebra, or Cherednik algebra, is an algebra containing the Hecke algebra of an affine Weyl group, given as the quotient of the group ring of a double affine brai
Hall algebra
In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups. It was first discussed by but forgotten until it was rediscover
Poisson algebra
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 derivation. Poisson algebras appear naturally
Associative algebra
In mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some f
*-algebra
In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has
Ternary commutator
In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by Also called the ternutator or alternating ternary sum, it is a special case of the for
Peirce decomposition
In ring theory, a Peirce decomposition /ˈpɜːrs/ is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements. The Peirce decomposition for associative algebras was introdu
Max-plus algebra
No description available.
Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multip
Algebra homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if A and B are algebras over a field (or commutative ring) K, it is a function such that for
Difference algebra
Difference algebra is a branch of mathematics concerned with the study of difference (or functional) equations from the algebraic point of view. Difference algebra is analogous to differential algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
Augmentation (algebra)
In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism , typically denoted by ε. An algebra together with an augmentation is called an augmented
Cellular algebra
In abstract algebra, a cellular algebra is a finite-dimensional associative algebra A with a distinguished cellular basis which is particularly well-adapted to studying the representation theory of A.
Finitely generated algebra
In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A over a field K where there exists a finite set of elements a1,...,an of A su
Rng (algebra)
In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existe
Frobenius theorem (real division algebras)
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the r
Algebra over a field
In 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 consisting of a set together wit
Normed algebra
In mathematics, a normed algebra A is an algebra over a field which has a sub-multiplicative norm: Some authors require it to have a multiplicative identity 1A such that ║1A║ = 1.
Colombeau algebra
In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is
Koszul duality
In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimp
List of algebras
This is a list of possibly nonassociative algebras. An algebra is a module, wherein you can also multiply two module elements. (The multiplication in the module is compatible with multiplication-by-sc
Pseudo-ring
In mathematics, and more specifically in abstract algebra, a pseudo-ring is one of the following variants of a ring: * A rng, i.e., a structure satisfying all the axioms of a ring except for the exis
Algebra representation
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be ma
Simple algebra (universal algebra)
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 with domain A is either injective or const
Tensor algebra
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 rank) with multiplication being the tensor product. It is the free algebra on V, i
Functional-theoretic algebra
Any vector space can be made into a unital associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionals. In general, it is a non-commutative algebr
Cluster algebra
Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky . A cluster algebra of rank n is an integral domain A, together with some subsets of size n called clusters whose u
Hurwitz quaternion order
The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with
Separable algebra
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 extension.
Affine Hecke algebra
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 for Macdonald polynomials.
Generalized Clifford algebra
In mathematics, a Generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these
Gerstenhaber algebra
In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combin
Hecke algebra of a finite group
The Hecke algebra of a finite group is the algebra spanned by the double cosets HgH of a subgroup H of a finite group G. It is a special case of a Hecke algebra of a locally compact group.
Tensor product of algebras
In mathematics, 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, the most common application of s
Finite algebra
In abstract algebra, an -algebra is finite if it is finitely generated as an -module. An -algebra can be thought as a homomorphism of rings , in this case is called a finite morphism if is a finite -a
Peak algebra
In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group Sn, studied by . It consists of the elements of the group algebra of the symmetric group whose
Exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exte
Group algebra of a locally compact group
In functional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra)
Leavitt path algebra
In mathematics, a Leavitt path algebra is a universal algebra constructed from a directed graph. The Leavitt path algebras generalize the and may also be considered as algebraic analogues of the graph
Free algebra
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 its elements may be described as "polynomials"
Topological algebra
In mathematics, a topological algebra is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Iwahori–Hecke algebra
In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group. Hecke algebras are quotients of the
Fourier algebra
Fourier and related algebras occur naturally in the harmonic analysis of locally compact groups. They play an important role in the duality theories of these groups. The Fourier–Stieltjes algebra and
Nilpotent algebra
In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n
Weyl algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form More precisely, let F be the underlying field
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 such that . The index set is usually the set of nonneg
Frobenius algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which g
Batalin–Vilkovisky formalism
In theoretical physics, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theorie
Partial group algebra
In mathematics, a partial group algebra is an associative algebra related to the of a group.
Derivative algebra (abstract algebra)
In abstract algebra, a derivative algebra is an algebraic structure of the signature where is a Boolean algebra and D is a unary operator, the derivative opera
Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Koszul algebra
In abstract algebra, a Koszul algebra is a graded -algebra over which the ground field has a linear minimal graded free resolution, i.e., there exists an exact sequence: Here, is the graded algebra wi
Symmetric algebra
In 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 is, in some sense, minimal for this property. H
Bracket algebra
In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolic representation of projective invariants. Given that L is a proper signed al
Azumaya algebra
In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case
Central simple algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A which is simple, and for which the center is exactly K. (N
Rota–Baxter algebra
In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathema