- algebra, higher algebra
- universal algebra
- monoid, semigroup, quasigroup
- nonassociative algebra
- associative unital algebra
- commutative algebra
- Lie algebra, Jordan algebra
- Leibniz algebra, pre-Lie algebra
- Poisson algebra, Frobenius algebra
- lattice, frame, quantale
- Boolean ring, Heyting algebra
- commutator, center
- monad, comonad
- distributive law

- group, normal subgroup
- action, Cayley's theorem
- centralizer, normalizer
- abelian group, cyclic group
- group extension, Galois extension
- algebraic group, formal group
- Lie group, quantum group

Given a commutative field $k$, an **affine algebra** is any finitely generated noetherian associative commutative unital $k$-algebra without nilpotent elements.

This term is limited to a small community of algebraists and may bring confusion if quoted outside of that context. The name is from the fact that the affine algebras are the coordinate rings of affine varieties; equivalently, an affine $k$-variety is precisely the maximal spectrum of an affine $k$-algebra.

Some people drop the condition of commutativity and talk about noncommutative affine algebras. This is even less standard as in the noncommutative context there is a further hesitation to drop noetherianess as well.

Yet another confusion may arise when people informally say affine algebra for an affine Lie algebra; of course, the latter, full, term is preferred (or “affine Kac-Moody algebra”).

category: algebra

Last revised on September 12, 2024 at 17:48:11. See the history of this page for a list of all contributions to it.