Graded algebra

Graded algebra

In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field, or more general R-algebra, in which there is a consistent notion of the weight of an element. The idea is that the weights of elements should add, when elements are multiplied. One has to allow the 'inconsistent' addition of elements of different weights, though. A formal definition follows.

Let G be an abelian group. A G-graded algebra A is an algebra with a direct sum decomposition

such that

An element of the ith subspace A_i is said to be a homogeneous, or a pure, element of degree i.

Important examples of graded algebras include the tensor algebra of a vector space V as well as the exterior algebra both of which are Z-graded.

Clifford algebras and superalgebras are examples of Z₂-graded algebras. Here the homogeneous elements are either even (degree 0) or odd (degree 1).

Graded algebras are also much used in commutative algebra and algebraic geometry, homological algebra and algebraic topology.