9. Algebras over Fields

Definition

Let $K$ be a field.

An algebra over $K$ is a vector space $A$ over $K$ equipped with an internal multiplication

$A\times A\to A, \qquad (x,y)\mapsto xy,$ that is bilinear.

Bilinear means:

$(\alpha x+\beta y)z=\alpha(xz)+\beta(yz),$ and

$z(\alpha x+\beta y)=\alpha(zx)+\beta(zy).$ Depending on context, one may additionally require the multiplication to be associative, commutative, or to have an identity element.

Central intuition

An algebra over a field is a vector space whose elements can also multiply each other.

So:

Structure	What you can do
Abelian group	add elements
Vector space	add elements and scale them
Algebra over $K$	add, scale, and multiply elements internally

The essential slogan is:

An algebra is a linear space with multiplication compatible with linearity.

Why bilinearity is the key

If multiplication were arbitrary, it would not interact well with vector-space structure.

Bilinearity says multiplication respects linear combinations in each variable.

For example:

$(x+y)z=xz+yz,$ $x(y+z)=xy+xz,$ $(\lambda x)y=\lambda(xy)=x(\lambda y).$ So multiplication can be computed by expanding across sums and pulling out scalars.

This is the vector-space analogue of distributivity in rings.

Why algebras are natural

Many important vector spaces have a natural product.

For example:

functions can be multiplied pointwise;
polynomials can be multiplied;
matrices can be multiplied;
linear operators can be composed;
differential forms can be multiplied by wedge product;
Lie algebra elements can be combined by a bracket.

In all these cases, the product is compatible with linear combinations.

Algebra as ring plus vector space

An algebra over $K$ can be seen as an object that is both:

a vector space over $K$;
a ring-like object under $+$ and multiplication;

with compatibility between scalar multiplication and internal multiplication.

This is why algebras sit at the intersection of ring theory and linear algebra.

Natural examples

The field itself

$K$ is an algebra over itself.

Polynomial algebra

$K[x]$ is an algebra over $K$.

It is a vector space with basis

$1,x,x^2,x^3,\dots$ and it also has polynomial multiplication.

Matrix algebra

$M_n(K)$ is an algebra over $K$.

It is usually non-commutative:

$AB\neq BA$ in general.

This is one of the most important examples because it shows that algebras can be linear and non-commutative at the same time.

Endomorphism algebra

If $V$ is a vector space, then

$\operatorname{End}(V)$ is the algebra of linear maps $V\to V$, with multiplication given by composition.

Function algebra

Functions $X\to K$ form an algebra under pointwise operations.

Associative, commutative, and non-associative algebras

The word “algebra” does not always mean the multiplication is associative.

Associative algebras

$(xy)z=x(yz).$ Examples: matrices, polynomials, functions.

Commutative algebras

$xy=yx.$ Examples: polynomial rings, function algebras.

Non-associative algebras

Examples include Lie algebras, where the product is a bracket

$[x,y]$ satisfying different laws such as antisymmetry and the Jacobi identity.

What algebras make possible

Algebras let us build expressions involving both linear combinations and products:

$x^2+3xy-2y^2,$ $xy-yx,$ $x^3+ax+b.$ Thus algebras are natural environments for polynomial expressions in abstract objects.

Place in the build-up

Algebras over fields combine two previous lines:

the vector-space line: addition plus scalar multiplication;
the ring line: addition plus internal multiplication.

So an algebra is:

a vector space whose vectors also multiply, with multiplication compatible with linear combinations.