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This article is about a particular kind of vector space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added For other uses of the term "algebra" see algebra (disambiguation). Algebra is a branch of Mathematics. Algebra may also mean Elementary algebra Abstract algebra

In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Mathematics, associativity is a property that a Binary operation can have They are thus special algebras. In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with

Contents

Definition

An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A → A (where the image of (x,y) is written as xy) such that the associative law holds:

The bilinearity of the multiplication can be expressed as

If A contains an identity element, i. e. an element 1 such that 1x = x1 = x for all x in A, then we call A a associative algebra with one or an unital (or unitary) associative algebra. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i Such an algebra is a ring, and contains all elements a of the field K by identification with a1. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

The dimension of the associative algebra A over the field K is its dimension as a K-vector space. In Mathematics, the dimension of a Vector space V is the cardinality (i

Modules

The preceding definition generalizes without any change to an algebra over a commutative ring K. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property Such a space is then a module, rather than a vector space, over K with a bilinear form. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars A unital R-algebra A can equivalently be defined as a ring A with a ring homomorphism R→A. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real For instance:

See algebra (ring theory) for more. In Mathematics, specifically in Ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the

Examples

Algebra homomorphisms

If A and B are associative algebras over the same field K, an algebra homomorphism h: A → B is a K-linear map which is also multiplicative in the sense that h(xy) = h(x) h(y) for all x, y in A. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that With this notion of morphism, the class of all associative algebras over K becomes a category. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

Take for example the algebra A of all real-valued continuous functions R → R, and B = R. Both are algebras over R, and the map which assigns to every continuous function f the number f(0) is an algebra homomorphism from A to B.

Associativity and the multiplication mapping

Associativity was defined above quantifying over all elements of A. It is possible to define associativity in a way that does not explicitly refer to elements. An algebra is defined as a map M (multiplication) on a vector space A:

M: A \times A \rightarrow A

An associative algebra is an algebra where the map M has the property

M \circ (\mbox {Id} \times M) = M \circ (M \times \mbox {Id})

Here, the symbol \circ refers to function composition, and Id : A → A is the identity map on A. In Mathematics, a composite function represents the application of one function to the results of another This article is about the Identity Map software design pattern

To see the equivalence of the definitions, we need only understand that each side of the above equation is a function that takes three arguments. For example, the left-hand side acts as

( M \circ (\mbox {Id} \times M)) (x,y,z) = M (x, M(y,z))

Similarly, a unital associative algebra can be defined in terms of a unit map

\eta: K \rightarrow A

which has the property

M \circ (\mbox {Id} \times \eta ) = s = M \circ (\eta \times \mbox {Id})

Here, the unit map η takes an element k in K to the element k1 in A, where 1 is the unit element of A. The map s is just plain-old scalar multiplication: s:K\times A \rightarrow A; thus, the above identity is sometimes written with Id standing in the place of s, with scalar multiplication being implicitly understood.

Coalgebras

An associative unitary algebra over K is based on a morphism A×A→A having 2 inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K→A identifying the scalar multiples of the multiplicative identity. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams which describe the algebra axioms; this defines the structure of a coalgebra. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Mathematics, coalgebras are structures that are dual to Unital Associative algebras The Axioms of unital associative algebras can

There is also an abstract notion of F-coalgebra. In Mathematics, specifically in Category theory, an F-coalgebra for an Endofunctor F: \mathbf{C}\longrightarrow \mathbf{C}

Representations

A representation of an algebra is a linear map ρ: A → gl(V) from A to the general linear algebra of some vector space (or module) V that preserves the multiplicative operation: that is, ρ(xy)=ρ(x)ρ(y). In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

Note, however, that there is no natural way of defining a tensor product of representations of associative algebras, without somehow imposing additional conditions. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector Here, by tensor product of representations, the usual meaning is intended: the result should be a linear representation on the product vector space. Imposing such additional structure typically leads to the idea of a Hopf algebra or a Lie algebra, as demonstrated below. In Mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( Unital associative algebra, a Coalgebra In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

Motivation for a Hopf algebra

Consider, for example, two representations \sigma:A\rightarrow gl(V) and \tau:A\rightarrow gl(W). One might try to form a tensor product representation \rho: x \mapsto \rho(x) = \sigma(x) \otimes \tau(x) according to how it acts on the product vector space, so that

\rho(x)(v \otimes w) = (\sigma(x)(v)) \otimes (\tau(x)(w)).

However, such a map would not be linear, since one would have

\rho(kx) = \sigma(kx) \otimes \tau(kx) = k\sigma(x) \otimes k\tau(x) = k^2 (\sigma(x) \otimes \tau(x)) = k^2 \rho(x)

for k ∈ K. One can rescue this attempt and restore linearity by imposing additional structure, by defining a map Δ: A → A × A, and defining the tensor product representation as

\rho = (\sigma\otimes \tau) \circ \Delta.

Here, Δ is a comultiplication. In Mathematics, coalgebras are structures that are dual to Unital Associative algebras The Axioms of unital associative algebras can The resulting structure is called a bialgebra. In Mathematics, a bialgebra over a field K is a structure which is both a Unital Associative algebra and a Coalgebra over To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be unital as well. Note that bialgebras leave multiplication and co-multiplication unrelated; thus it is common to relate the two (by defining an antipode), thus creating a Hopf algebra. In Mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( Unital associative algebra, a Coalgebra

Motivation for a Lie algebra

One can try to be more clever in defining a tensor product. Consider, for example,

x \mapsto \rho (x) = \sigma(x) \otimes \mbox{Id}_W + \mbox{Id}_V \otimes \tau(x)

so that the action on the tensor product space is given by

\rho(x) (v \otimes w) = (\sigma(x) v)\otimes w + v \otimes (\tau(x) w) .

This map is clearly linear in x, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication:

\rho(xy) = \sigma(x) \sigma(y) \otimes \mbox{Id}_W + \mbox{Id}_V \otimes \tau(x) \tau(y).

But, in general, this does not equal

\rho(x)\rho(y) = \sigma(x) \sigma(y) \otimes \mbox{Id}_W + \sigma(x) \otimes \tau(y) + \sigma(y) \otimes \tau(x) + \mbox{Id}_V \otimes \tau(x) \tau(y).

Equality would hold if the product xy were antisymmetric (if the product were the Lie bracket, that is, xy \equiv M(x,y) = [x,y]), thus turning the associative algebra into a Lie algebra. Lie bracket can refer to Lie algebra Lie bracket of vector fields In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

References

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