Algebra over a field

An algebra over a field $k$ is a vector space $A$ over $k$ equipped with a bilinear multiplication $A\times A\to A$.

A $k$-algebra is finitely generated when finitely many elements generate it under addition, multiplication, and scalar multiplication. Equivalently, it is a quotient of a polynomial ring $k[x_1,\dots ,x_n]$.

For a vector space with quadratic form $Q$, its Clifford algebra is generated by vectors subject to $v^2=Q(v)1$. In a basis this is equivalently ${\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }=2{\eta }^{\mu \nu }1$.

Gamma matrices give a matrix representation of the Clifford algebra of spacetime.

A Hopf algebra is a compatible algebra and coalgebra equipped with a counit and antipode. Coordinate rings of affine algebraic groups are commutative Hopf algebras.

An associative algebra is an algebra over a field whose multiplication satisfies $(ab)c=a(bc)$. Unless stated otherwise, the algebras considered here have a multiplicative identity.

The group algebra $k[G]$ is the vector space with basis $G$ and multiplication obtained by extending the group law $k$-bilinearly.

A left ideal $L$ of an associative algebra $A$ is a vector subspace satisfying $AL\subseteq L$. Equivalently, it is a submodule of the left regular $A$-module.

A finite-dimensional associative algebra is semisimple when its left regular module is a direct sum of simple modules. Equivalently, every left ideal is a direct summand. In particular, a semisimple algebra has no nonzero square-zero left ideal: if $L^2=0$ and a module projection $A\to L$ exists, its value $e$ at $1$ satisfies $e\in L$ and $e^2=e$, forcing $e=0$.

If an algebra is defined over a parameter ring $R$, a ring homomorphism $R\to k$ produces a specialized $k$-algebra by extension of scalars. Defining relations specialize by replacing every parameter by its image in $k$.