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List Of Algebras
{{short description, None 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-scalars from the base ring). * Akivis algebra * Algebra for a monad * Albert algebra * Alternative algebra * Azumaya algebra * Banach algebra * Birman–Wenzl algebra * Boolean algebra * Borcherds algebra * Brauer algebra * C*-algebra * Central simple algebra * Clifford algebra * Cluster algebra * Dendriform algebra * Differential graded algebra * Differential graded Lie algebra * Exterior algebra * F-algebra * Filtered algebra * Flexible algebra * Freudenthal algebra * Genetic algebra * Geometric algebra * Gerstenhaber algebra * Graded algebra * Griess algebra * Group algebra * Group algebra of a locally compact group * Hall algebra * Hecke algebra of a locally compact group * Heyting algebra * Hopf algebra * Hurwitz algebra * Hypercomplex alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonassociative Algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if it is a vector space over ''K'' and is equipped with a ''K''- bilinear binary multiplication operation ''A'' × ''A'' → ''A'' which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (''ab'')(''cd''), (''a''(''bc''))''d'' and ''a''(''b''(''cd'')) may all yield different answers. While this use of ''non-associative'' means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluster Algebra
Cluster algebras are a class of commutative rings introduced by . A cluster algebra of rank ''n'' is an integral domain ''A'', together with some subsets of size ''n'' called clusters whose union generates the algebra ''A'' and which satisfy various conditions. Definitions Suppose that ''F'' is an integral domain, such as the field Q(''x''1,...,''x''''n'') of rational functions in ''n'' variables over the rational numbers Q. A cluster of rank ''n'' consists of a set of ''n'' elements of ''F'', usually assumed to be an algebraically independent set of generators of a field extension ''F''. A seed consists of a cluster of ''F'', together with an exchange matrix ''B'' with integer entries ''b''''x'',''y'' indexed by pairs of elements ''x'', ''y'' of the cluster. The matrix is sometimes assumed to be skew-symmetric, so that ''b''''x'',''y'' = –''b''''y'',''x'' for all ''x'' and ''y''. More generally the matrix might be skew-symmetrizable, meaning there are positive integers '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Ring
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 module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Definition Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Griess Algebra
In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 246320597611213317192329314147 ... ''M'' as its automorphism group. It is named after mathematician R. L. Griess, who constructed it in 1980 and subsequently used it in 1982 to construct ''M''. The Monster fixes (vectorwise) a 1-space in this algebra and acts absolutely irreducibly on the 196883-dimensional orthogonal complement of this 1-space. (The Monster preserves the standard inner product on the 196884-space.) Griess's construction was later simplified by Jacques Tits and John H. Conway. The Griess algebra is the same as the degree 2 piece of the monster vertex algebra, and the Griess product is one of the vertex alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Algebra
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 such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded \Z-algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. First properties Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 combines the structures of a supercommutative ring and a graded Lie superalgebra. It is used in the Batalin–Vilkovisky formalism. It appears also in the generalization of Hamiltonian formalism known as the De Donder–Weyl theory as the algebra of generalized Poisson brackets defined on differential forms. Definition A Gerstenhaber algebra is a graded-commutative algebra with a Lie bracket of degree −1 satisfying the Poisson identity. Everything is understood to satisfy the usual superalgebra sign conventions. More precisely, the algebra has two products, one written as ordinary multiplication and one written as and a Z-grading called degree (in theoretical physics sometimes called ghost number). The degree of an element ''a'' is denoted by , ''a'', . These satisfy the identiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions. The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genetic Algebra
In mathematical genetics, a genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weighted algebra). The study of these algebras was started by . In applications to genetics, these algebras often have a basis corresponding to the genetically different gametes, and the structure constant of the algebra encode the probabilities of producing offspring of various types. The laws of inheritance are then encoded as algebraic properties of the algebra. For surveys of genetic algebras see , and . Baric algebras Baric algebras (or weighted algebras) were introduced by . A baric algebra over a field ''K'' is a possibly non-associative algebra over ''K'' together with a homomorphism ''w'', called the weight, from the algebra to ''K''. Ber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freudenthal Algebra
In algebra, Freudenthal algebras are certain Jordan algebras constructed from composition algebras. Definition Suppose that ''C'' is a composition algebra over a field ''F'' and ''a'' is a diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ... in GL''n''(''F''). A reduced Freudenthal algebra is defined to be a Jordan algebra equal to the set of 3 by 3 matrices ''X'' over ''C'' such that T''a''=''aX''. A Freudenthal algebra is any twisted form of a reduced Freudental algebra. References * * Non-associative algebras {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flexible Algebra
In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: : a \bullet \left(b \bullet a\right) = \left(a \bullet b\right) \bullet a for any two elements ''a'' and ''b'' of the set. A magma (that is, a set equipped with a binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, a nonassociative algebra is flexible if its multiplication operator is flexible. Every commutative or associative operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for the multiplication of sedenions, which are not even alternative. In 1954, Richard D. Schafer examined the algebras generated by the Cayley–Dickson process over a field and showed that they satisfy the flexible identity.Richard D. Schafer (1954) “On the algebras formed by the Cayley-Dickson process”, American Journal of Mathematics 76: 435 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. A filtered algebra over the field k is an algebra (A,\cdot) over k that has an increasing sequence \ \subseteq F_0 \subseteq F_1 \subseteq \cdots \subseteq F_i \subseteq \cdots \subseteq A of subspaces of A such that :A=\bigcup_ F_ and that is compatible with the multiplication in the following sense: : \forall m,n \in \mathbb,\quad F_m\cdot F_n\subseteq F_. Associated graded algebra In general there is the following construction that produces a graded algebra out of a filtered algebra. If A is a filtered algebra then the ''associated graded algebra'' \mathcal(A) is defined as follows: The multiplication is well-defined and endows \mathcal(A) with the structure of a graded algebra, with gradation \_. Furthermore if A is associative then so is \mathcal(A). Also if A is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F-algebra
In mathematics, specifically in category theory, ''F''-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor ''F'', the ''signature''. ''F''-algebras can also be used to represent data structures used in programming, such as lists and trees. The main related concepts are initial ''F''-algebras which may serve to encapsulate the induction principle, and the dual construction ''F''-coalgebras. Definition If C is a category, and F : C \rightarrow C is an endofunctor of C, then an F-algebra is a tuple (A, \alpha), where A is an object of C and \alpha is a C- morphism F(A) \rightarrow A. The object A is called the ''carrier'' of the algebra. When it is permissible from context, algebras are often referred to by their carrier only instead of the tuple. A homomorphism from an F-al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
