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Encyclopedia > Associative 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. They are thus special algebras. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... 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 to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...

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 AA (where the image of (x,y) is written as xy) such that the associative law holds: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

  • (x y) z = x (y z) for all x, y and z in A.

The bilinearity of the multiplication can be expressed as

  • (x + y) z = x z + y z    for all x, y, z in A,
  • x (y + z) = x y + x z    for all x, y, z in A,
  • a (x y) = (a x) y = x (a y)    for all x, y in A and a in K.

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. Such an algebra is a ring, and contains all elements a of the field K by identification with a1. In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...


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. Such a space is then a module, rather than a vector space, over K with a bilinear form. A unital R-algebra A can equivalently be defined as a ring A with a ring homomorphism RA. For instance: In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... 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 to lie in a field, the scalars may lie in an arbitrary ring. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...

  • The n-by-n matrices with integer entries form a unital associative algebra over the integers.
  • The polynomials with coefficients in the ring Z/nZ, the integers modulo n, form a unital associative algebra over Z/nZ.

See algebra (ring theory) for more. The integers are commonly denoted by the above symbol. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... 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 base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ...


Examples

  • The square n-by-n matrices with entries from the field K form a unitary associative algebra over K.
  • The complex numbers form a 2-dimensional unitary associative algebra over the real numbers.
  • The quaternions form a 4-dimensional unitary associative algebra over the reals (but not an algebra over the complex numbers, since complex numbers don't commute with quaternions).
  • The polynomials with real coefficients form a unitary associative algebra over the reals.
  • Given any Banach space X, the continuous linear operators A : XX form a unitary associative algebra (using composition of operators as multiplication); this is in fact a Banach algebra.
  • Given any topological space X, the continuous real- (or complex-) valued functions on X form a real (or complex) unitary associative algebra; here we add and multiply functions pointwise.
  • An example of a non-unitary associative algebra is given by the set of all functions f: RR whose limit as x nears infinity is zero.
  • The Clifford algebras are useful in geometry and physics.
  • Incidence algebras of locally finite partially ordered sets are unitary associative algebras considered in combinatorics.

In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... Clifford algebras are a type of associative algebra in mathematics. ... Table of Geometry, from the 1728 Cyclopaedia. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe. ... In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval [a, b] = {x : a ≤ x ≤ b} within it is finite. ... In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ...

Algebra homomorphisms

If A and B are associative algebras over the same field K, an algebra homomorphism h: AB 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. With this notion of morphism, the class of all associative algebras over K becomes a category. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...


Take for example the algebra A of all real-valued continuous functions RR, 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.


Index-free notation

In the above definition of an associative algebra, the definition of associativity was made with regard to all of the elements of A. It is sometimes more convenient to have a definition of associativity that does not need to refer to the elements of A. This can be done as follows. 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 functional composition, and Id is the identity map: Id(x) = x for all x in A. 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:Ktimes 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×AA having 2 inputs (multiplicator and multiplicand) and one output (product), as well as a morphism KA identifying the scalar multiples of the multiplicative identity. 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 mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... This article does not adequately cite its references or sources. ... In mathematics, coalgebras are structures that are in a certain sense dual to the unital associative algebras. ...


There is also an abstract notion of F-coalgebra. In mathematics, specifically in category theory, an -coalgebra for an endofunctor is an object of together with a -morphism . In this sense F-coalgebras are dual to F-algebras. ...


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). Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...


Note, however, that there is no natural way of defining a tensor product of representations of associative algebras, without somehow imposing additional conditions. 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, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... In mathematics, a Hopf algebra, named after Heinz Hopf, is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...


Motivation for a Hopf algebra

Consider, for example, two representations sigma:Arightarrow gl(V) and tau:Arightarrow 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) = ksigma(x) otimes ktau(x) = k^2 (sigma(x) otimes tau(x)) = k^2 rho(x)

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

rho = (sigmaotimes tau) circ Delta.

Here, Δ is a comultiplication. The resulting structure is called a bialgebra. 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, coalgebras are structures that are in a certain sense dual to the unital associative algebras. ... In mathematics, a bialgebra over a field K is a structure which is both a unital associative algebra and a coalgebra over K, such that the comultiplication and the counit are both unital algebra homomorphisms. ... In mathematics, a Hopf algebra, named after Heinz Hopf, is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ...


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. A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...


References

  • Ross Street, Quantum Groups: an entrée to modern algebra (1998). (Provides a good overview of index-free notation)

  Results from FactBites:
 
PlanetMath: non-associative algebra (176 words)
A non-associative algebra is an algebra in which the assumption of multiplicative associativity is dropped.
In much of the literature concerning non-associative algebras, where the meaning of a “non-associative algebra” is clear, the word “non-associative” is dropped for simplicity and clarity.
Lie algebras and Jordan algebras are two famous examples of non-associative algebras that are not associative.
PlanetMath: non-associative algebra (174 words)
A non-associative algebra is an algebra in which the assumption of multiplicative associativity is dropped.
In much of the literature concerning non-associative algebras, where the meaning of a “non-associative algebra” is clear, the word “non-associative” is dropped for simplicity and clarity.
Lie algebras and Jordan algebras are two famous examples of non-associative algebras that are not associative.
  More results at FactBites »

 
 

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