Associative algebra

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In mathematics, an associative algebra A is an ring (not necessarily unital) that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

For example, a ring of square matrices over a field K is an associative K algebra. More generally, given a ring S with center C, S is an associative C algebra.

In some areas of mathematics, associative algebras are typically assumed to have a multiplicative unit, denoted 1. To make this extra assumption clear, these associative algebras are called unital algebras. Additionally, some authors demand that all rings be unital; in this article, the word "ring" is intended to refer to potentially non-unital rings as well.

Formal definition

Let R be a fixed commutative ring. An associative 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 ring multiplication is R-bilinear:

${\displaystyle r\cdot (xy)=(r\cdot x)y=x(r\cdot y)}$

for all rR and x, yA. We say A is unital if it contains an element 1 such that

${\displaystyle 1x=x=x1}$

for all xA. Note that such an element 1 must be unique if it exists at all.

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

From R-modules

Starting with an R-module A, we get an associative R-algebra by equipping A with an R-bilinear mapping A × AA such that

${\displaystyle x(yz)=(xy)z\,}$

for all x, y, and z in A. This R-bilinear mapping then gives A the structure of a ring and an associative R-algebra. Every associative R-algebra arises this way.

Moreover, the algebra A built this way will be unital if and only if there exists an element 1 of A such that every element x of A satisfies 1x = x1 = x. This definition is equivalent to the statement that a unital associative R-algebra is a monoid in R-Mod (the monoidal category of R-modules).

From rings

Starting with a ring A, we get a unital associative R-algebra by providing a ring homomorphism ${\displaystyle \eta \colon R\to A}$ whose image lies in the center of A. The algebra A can then be thought of as an R-module by defining

${\displaystyle r\cdot x=\eta (r)x}$

for all rR and xA.

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

Algebra homomorphisms

A homomorphism between two associative R-algebras is an R-linear ring homomorphism. Explicitly, ${\displaystyle \phi :A_{1}\to A_{2}}$ is an associative algebra homomorphism if

${\displaystyle \phi (r\cdot x)=r\cdot \phi (x)}$
${\displaystyle \phi (x+y)=\phi (x)+\phi (y)\,}$
${\displaystyle \phi (xy)=\phi (x)\phi (y)\,}$

For a homomorphism of unital associative R-algebras, we also demand that

${\displaystyle \phi (1)=1\,}$

The class of all unital associative 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

• Any (unital) ring A can be considered as a unital Z-algebra in a unique way. 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 unital Z-algebras are equivalent concepts, in the same way that abelian groups and Z-modules are equivalent.
• Any ring of characteristic n is a (Z/nZ)-algebra in the same way.
• Given an R-module M, the endomorphism ring of M, denoted EndR(M) is an R-algebra by defining (r·φ)(x) = r·φ(x).
• Any ring of matrices 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, free R-module.
• The square n-by-n matrices with entries from the field K form a unital associative algebra over K. In particular, the 2 × 2 real matrices form an associative algebra useful in plane mapping.
• The complex numbers form a 2-dimensional unital associative algebra over the real numbers.
• The quaternions form a 4-dimensional unital associative algebra over the reals (but not an algebra over the complex numbers, since if complex numbers are treated as a subset of the quaternions, complex numbers and quaternions do not commute).
• The polynomials with real coefficients form a unital associative algebra over the reals.
• Every polynomial ring R[x1, ..., xn] is a commutative R-algebra. In fact, this is the free commutative R-algebra on the set {x1, ..., xn}.
• The free R-algebra on a set E is an algebra of polynomials with coefficients in R and noncommuting indeterminates taken from the set E.
• The tensor algebra of an R-module is naturally an R-algebra. The same is true for quotients such as the exterior and symmetric algebras. Categorically speaking, the functor which maps an R-module to its tensor algebra is left adjoint to the functor which sends an R-algebra to its underlying R-module (forgetting the ring structure).
• Given a commutative ring R and any ring A the tensor product RZA can be given the structure of an R-algebra by defining r·(sa) = (rsa). The functor which sends A to RZA is left adjoint to the functor which sends an R-algebra to its underlying ring (forgetting the module structure).

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.

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 vector space A with a bilinear map

${\displaystyle M:A\times A\rightarrow A}$

(the multiplication map). An associative algebra is an algebra where the map M has the property

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

Here, the symbol ${\displaystyle \circ }$ refers to function composition, and Id : A → A is the identity map on 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

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

Similarly, a unital associative algebra can be defined as a vector space A endowed with a map M as above and, additionally, a linear map

${\displaystyle \eta :K\rightarrow A}$

(the unit map) which has the properties

${\displaystyle M\circ ({\mbox{Id}}\times \eta )=s;\ M\circ (\eta \times {\mbox{Id}})=t}$

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 t is just plain-old scalar multiplication: ${\displaystyle t:K\times A\rightarrow A,\ \left(k,a\right)\mapsto ka}$; the map s is similar: ${\displaystyle s:A\times K\rightarrow A,\ \left(a,k\right)\mapsto ka}$.

Coalgebras

{{#invoke:main|main}} An associative unital 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 unital 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. This is vaguely related to the notion of coalgebra discussed above.

Representations

{{#invoke:main|main}} A representation of a unital algebra A is a unital algebra homomorphism ρ: A → End(V) from A to the endomorphism algebra of some vector space (or module) V. The property of ρ being a unital 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 ${\displaystyle \sigma :A\rightarrow \mathrm {End} (V)}$ and ${\displaystyle \tau :A\rightarrow \mathrm {End} (W)}$. One might try to form a tensor product representation ${\displaystyle \rho :x\mapsto \sigma (x)\otimes \tau (x)}$ according to how it acts on the product vector space, so that

${\displaystyle \rho (x)(v\otimes w)=(\sigma (x)(v))\otimes (\tau (x)(w)).}$

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

${\displaystyle \rho (kx)=\sigma (kx)\otimes \tau (kx)=k\sigma (x)\otimes k\tau (x)=k^{2}(\sigma (x)\otimes \tau (x))=k^{2}\rho (x)}$

for kK. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism Δ: AAA, and defining the tensor product representation as

${\displaystyle \rho =(\sigma \otimes \tau )\circ \Delta .}$

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

{{#invoke:see also|seealso}} One can try to be more clever in defining a tensor product. Consider, for example,

${\displaystyle x\mapsto \rho (x)=\sigma (x)\otimes {\mbox{Id}}_{W}+{\mbox{Id}}_{V}\otimes \tau (x)}$

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

${\displaystyle \rho (x)(v\otimes w)=(\sigma (x)v)\otimes w+v\otimes (\tau (x)w)}$.

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:

${\displaystyle \rho (xy)=\sigma (x)\sigma (y)\otimes {\mbox{Id}}_{W}+{\mbox{Id}}_{V}\otimes \tau (x)\tau (y)}$.

But, in general, this does not equal

${\displaystyle \rho (x)\rho (y)=\sigma (x)\sigma (y)\otimes {\mbox{Id}}_{W}+\sigma (x)\otimes \tau (y)+\sigma (y)\otimes \tau (x)+{\mbox{Id}}_{V}\otimes \tau (x)\tau (y)}$.

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).