Associative algebra

This article is about a particular kind of algebra over a commutative ring. For other uses of the term "algebra", see Algebra (disambiguation).

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.

In this article associative algebras are assumed to have a multiplicative unit, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.

Many authors consider the more general concept of an associative algebra over a commutative ring R, instead of a field: An R-algebra is an R-module with an associative R-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if S is any ring with center C, then S is an associative C-algebra.

Definition

Let R be a fixed commutative ring (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is an additive abelian group A which has the structure of both a ring and an R-module in such a way that the scalar multiplication satisfies

r\cdot(xy) = (r\cdot x)y = x(r\cdot y)

for all rR and x, yA. Furthermore, A is assumed to be unital, which is to say it contains an element 1 such that

1 x = x = x 1

for all xA. Note that such an element 1 must be unique.

In other words, A is an R-module together with (1) an R-bilinear map A × AA, called the multiplication, and (2) the multiplicative identity, such that the multiplication is associative:

x(yz) = (xy)z\,

for all x, y, and z in A. (Technical note: the multiplicative identity is a datum,[1] while associativity is a property. By the uniqueness of the multiplicative identity, "unitarity" is often treated like a property.) If one drops the requirement for the associativity, then one obtains a non-associative algebra.

If A itself is commutative (as a ring) then it is called a commutative R-algebra.

As a monoid object in the category of modules

The definition is equivalent to saying that a unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, an associative algebra is obtained by replacing the category of abelian groups with the category of modules.

This reinterpretation of the definition is convenient for further generalization since one does not need to refer to elements of an algebra A explicitly. For example, the associativity can be expressed as follows. By the universal property of a tensor product of modules, the multiplication (the R-bilinear map) corresponds to a unique R-linear map

m: A \otimes_R A \to A.

The associativity then refers to the identity:

m \circ (\operatorname {id} \otimes m) = m \circ (m \otimes \operatorname {id}).

From ring homomorphisms

An associative algebra amounts to a ring homomorphism whose image lies in the center. Indeed, starting with a ring A and a ring homomorphism \eta\colon R \to A whose image lies in the center of A, we can make A an R-algebra by defining

r\cdot x = \eta(r)x

for all rR and xA. If A is an R-algebra, taking x = 1, the same formula in turn defines a ring homomorphism \eta\colon R \to A whose image lies in the center.

If A is commutative then the center of A is equal to A, so that a commutative R-algebra can be defined simply as a homomorphism \eta\colon R \to A of commutative rings.

The ring homomorphism η appearing in the above is often called a structure map. In the commutative case, one can consider the category whose objects are ring homomorphisms RA; i.e., commutative R-algebras and whose morphisms are ring homomorphisms AA' that are under R; i.e., RAA' is RA' (i.e., the coslice category of the category of commutative rings under R.) The prime spectrum functor Spec then determines an anti-equivalence of this category to the category of affine schemes over Spec R. (How to weaken the commutativity assumption is a subject matter of noncommutative algebraic geometry and, more recently, of derived algebraic geometry.)

Algebra homomorphisms

A homomorphism between two R-algebras is an R-linear ring homomorphism. Explicitly, \phi : A_1 \to A_2 is an associative algebra homomorphism if

\phi(r\cdot x) = r\cdot \phi(x)
\phi(x+y) = \phi(x)+\phi(y)\,
\phi(xy) = \phi(x)\phi(y)\,
\phi(1) = 1\,

The class of all R-algebras together with algebra homomorphisms between them form a category, sometimes denoted R-Alg.

The subcategory of commutative R-algebras can be characterized as the coslice category R/CRing where CRing is the category of commutative rings.

Examples

The most basic example is a ring itself; it is an algebra over its center or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics.

Algebra

Representation theory

Analysis

Geometry and combinatorics

Constructions

Subalgebras
A subalgebra of an R-algebra A is a subset of A which is both a subring and a submodule of A. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of A.
Quotient algebras
Let A be an R-algebra. Any ring-theoretic ideal I in A is automatically an R-module since r·x = (r1A)x. This gives the quotient ring A/I the structure of an R-module and, in fact, an R-algebra. It follows that any ring homomorphic image of A is also an R-algebra.
Direct products
The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication.
Free products
One can form a free product of R-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of R-algebras.
Tensor products
The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras for more details.

Coalgebras

Main article: Coalgebra

An associative algebra over K is given by a K-vector space A endowed with a bilinear map 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. If the bilinear map A×AA is reinterpreted as a linear map (i. e., morphism in the category of K-vector spaces) AAA (by the universal property of the tensor product), then we can view an associative algebra over K as a K-vector space A endowed with two morphisms (one of the form AAA and one of the form KA) satisfying certain conditions which boil down to the algebra axioms. 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.

There is also an abstract notion of F-coalgebra, where F is a functor. This is vaguely related to the notion of coalgebra discussed above.

Representations

A representation of an algebra A is an algebra homomorphism ρ: A → End(V) from A to the endomorphism algebra of some vector space (or module) V. The property of ρ being an algebra homomorphism means that ρ preserves the multiplicative operation (that is, ρ(xy)=ρ(x)ρ(y) for all x and y in A), and that ρ sends the unity of A to the unity of End(V) (that is, to the identity endomorphism of V).

If A and B are two algebras, and ρ: A → End(V) and τ: B → End(W) are two representations, then it is easy to define a (canonical) representation A B → End(V W) of the tensor product algebra A B on the vector space V W. Note, however, that there is no natural way of defining a tensor product of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by tensor product of representations, the usual meaning is intended: the result should be a linear representation of the same algebra 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.

Motivation for a Hopf algebra

Consider, for example, two representations \sigma:A\rightarrow \mathrm{End}(V) and \tau:A\rightarrow \mathrm{End}(W). One might try to form a tensor product representation \rho: x \mapsto \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 kK. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism Δ: AA A, and defining the tensor product representation as

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

Such a homomorphism Δ is called a comultiplication if it satisfies certain axioms. 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 co-unital as well. A Hopf algebra is a bialgebra with an additional piece of structure (the so-called antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups).

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).

This shows that this definition of a tensor product is too naive. It can be used, however, to define the tensor product of two representations of a Lie algebra (rather than of an associative algebra).

Non-unital algebras

Some authors use the term "associative algebra" to refer to structures with do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital.

An example of a non-unital associative algebra is given by the set of all functions f: RR whose limit as x nears infinity is zero.

See also

Notes

  1. Put in another way, there is the forgetful functor from the category of unital associative algebras to the category of possibly non-unital associative algebras.

References

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