Associative algebra

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

In this article associative algebras are assumed to have a multiplicative identity, 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 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 R, instead of a field: An R-algebra is an  with an associative R-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if S is any ring with C, then S is an associative C-algebra.

Definition
Let R be a fixed (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is an additive A which has the structure of both a  and an  in such a way that the  satisfies
 * $$r\cdot(xy) = (r\cdot x)y = x(r\cdot y)$$

for all r ∈ R and x, y ∈ A. 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 x ∈ A. Note that such an element 1 must be unique.

In other words, A is an R-module together with (1) an A &times; A → A, 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, 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.

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 in  (the  of R-modules). By definition, a ring is a monoid object in the ; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the.

Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra A. For example, the associativity can be expressed as follows. By the universal property of a, 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 whose image lies in the. Indeed, starting with a ring A and a ring homomorphism $$\eta\colon R \to A$$ whose image lies in the of A, we can make A an R-algebra by defining
 * $$r\cdot x = \eta(r)x$$

for all r ∈ R and x ∈ A. 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 ring is commutative then it equals its center, so that a commutative R-algebra can be defined simply as a commutative ring A together with a commutative ring homomorphism $$\eta\colon R \to A$$.

The ring homomorphism η appearing in the above is often called a. In the commutative case, one can consider the category whose objects are ring homomorphisms R → A; i.e., commutative R-algebras and whose morphisms are ring homomorphisms A → A that are under R; i.e., R → A → A is R → A' (i.e., the of the category of commutative rings under R.)  The  functor Spec then determines an anti-equivalence of this category to the category of s over Spec R.

How to weaken the commutativity assumption is a subject matter of and, more recently, of. See also:.

Algebra homomorphisms
A between two R-algebras is an. Explicitly, $$\varphi : A_1 \to A_2$$ is an associative algebra homomorphism if
 * $$\varphi(r\cdot x) = r\cdot \varphi(x)$$
 * $$\varphi(x+y) = \varphi(x)+\varphi(y)\,$$
 * $$\varphi(xy) = \varphi(x)\varphi(y)\,$$
 * $$\varphi(1) = 1\,$$

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

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

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

 * Any ring A can be considered as a Z-algebra. The unique ring homomorphism from Z to A is determined by the fact that it must send 1 to the identity in A. Therefore, rings and Z-algebras are equivalent concepts, in the same way that s and Z-modules are equivalent.
 * Any ring of n is a (Z/nZ)-algebra in the same way.
 * Given an R-module M, the of M, denoted EndR(M) is an R-algebra by defining (r·φ)(x) = r·φ(x).
 * Any ring of with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication. This coincides with the previous example when M is a finitely-generated,  R-module.
 * The square n-by-n with entries from the field K form an associative algebra over K. In particular, the  form an associative algebra useful in plane mapping.
 * The s form a 2-dimensional associative algebra over the s.
 * The s form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions).
 * The s with real coefficients form an associative algebra over the reals.
 * Every R[x1, ..., xn] is a commutative R-algebra. In fact, this is the free commutative R-algebra on the set {x1, ..., xn}.
 * The on a set E is an algebra of 'polynomials' with coefficients in R and noncommuting indeterminates taken from the set E.
 * The of an R-module is naturally an R-algebra. The same is true for quotients such as the  and s. Categorically speaking, the  that maps an R-module to its tensor algebra is  to the functor that sends an R-algebra to its underlying R-module (forgetting the multiplicative structure).
 * The following ring is used in the theory of s. Given a commutative ring A, let $$G(A) = 1 + tA[\![t]\!],$$ the set of formal power series with constant term 1. It is an abelian group with the group operation that is the multiplication of power series. It is then a ring with the multiplication, denoted by $$\circ$$, such that $$(1 + at) \circ (1 + bt) = 1 + abt,$$ determined by this condition and the ring axioms. The additive identity is 1 and the multiplicative identity is $$1 + t$$. Then $$A$$ has a canonical structure of a $$G(A)$$-algebra given by the ring homomorphism
 * $$\begin{cases} G(A) \to A \\ 1 + \sum_{i > 0} a_i t^i \mapsto a_1 \end{cases}$$
 * On the other hand, if A is a λ-ring, then there is a ring homomorphism
 * $$\begin{cases} A \to G(A) \\ a \mapsto 1 + \sum_{i > 0} \lambda^i(a)t^i \end{cases}$$
 * giving $$G(A)$$ a structure of an A-algebra.

Representation theory

 * The of a Lie algebra is an associative algebra that can be used to study the given Lie algebra.
 * If G is a group and R is a commutative ring, the set of all functions from G to R with finite support form an R-algebra with the convolution as multiplication. It is called the of G. The construction is the starting point for the application to the study of (discrete) groups.
 * If G is an (e.g., semisimple ), then the  of G is the  A corresponding to G. Many structures of G translate to those of A.

Analysis

 * Given any X, the  s A : X → X form an associative algebra (using composition of operators as multiplication); this is a.
 * Given any X, the continuous real- or complex-valued functions on X form a real or complex associative algebra; here the functions are added and multiplied pointwise.
 * The set of s defined on the (&Omega;, F, (Ft)t &ge; 0, P) forms a ring under.
 * The
 * An

Geometry and combinatorics

 * The s, which are useful in and.
 * s of s are associative algebras considered in.

Constructions

 * Subalgebras: A subalgebra of an R-algebra A is a subset of A which is both a and a  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 I in A is automatically an R-module since r·x = (r1A)x. This gives the  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 of R-algebras in a manner similar to the free product of groups. The free product is the  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 for more details. Given a commutative ring R and any ring A the  R⊗ZA can be given the structure of an R-algebra by defining r·(s⊗a) = (rs⊗a). The functor which sends A to R⊗ZA is  to the functor which sends an R-algebra to its underlying ring (forgetting the module structure). See also:.

Coalgebras
An associative algebra over K is given by a K-vector space A endowed with a bilinear map A&times;A → A having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K → A identifying the scalar multiples of the multiplicative identity. If the bilinear map A&times;A → A is reinterpreted as a linear map (i. e., in the category of K-vector spaces) A&otimes;A → A (by the ), then we can view an associative algebra over K as a K-vector space A endowed with two morphisms (one of the form A&otimes;A → A and one of the form K → A) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using by reversing all arrows in the s that describe the algebra s; this defines the structure of a.

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

Representations
A 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 unit of A to the unit of End(V) (that is, to the identity endomorphism of V).

If A and B are two algebras, and ρ: A → End(V) and &tau;: B → End(W) are two representations, then there is a (canonical) representation A $$\otimes$$ B → End(V $$\otimes$$ W) of the tensor product algebra A $$\otimes$$ B on the vector space V $$\otimes$$ W. However, there is no natural way of defining a 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 , 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 or a, 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 k ∈ K. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism Δ: A → A &otimes; A, and defining the tensor product representation as


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

Such a homomorphism Δ is called a if it satisfies certain axioms. The resulting structure is called a. 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 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; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a.

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

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

Another example is the vector space of continuous periodic functions, together with the.