Graded algebra

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In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading.

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Graded algebra

A graded algebra A is an algebra that can be written as a direct sum

<math>A = \bigoplus_{n\in N}A_i <math>

such that

<math> A_m A_n \subseteq A_{m + n}. <math>

A graded algebra is a special case of a graded vector space. Elements of <math>A_n<math> are known as homogeneous elements of degree n.

Examples of graded algebras are common in mathematics:

Graded algebras are much used in commutative algebra and algebraic geometry, homological algebra and algebraic topology. One example is the close relationship between homogeneous polynomials and projective varieties.

G-graded algebra

We can generalize the definition of a graded algebra to an arbitrary monoid as an index set. Let G be an monoid. A G-graded algebra A is an algebra with a direct sum decomposition

<math>A = \bigoplus_{i\in G}A_i <math>

such that

<math> A_i A_j \subseteq A_{i \cdot j} <math>

An element of the ith subspace Ai is said to be a homogeneous (or pure) element of degree i.

(If we don't require that the ring has an identity element, we can extend the definition from monoids to semigroups.

Examples of G-graded algebras include:

  • The group ring of a group is naturally graded by that group; similarly, monoid rings are graded by the corresponding monoid.

Category theoretically, a G-graded algebra A is an object in the category of G-graded vector spaces together with a morphism <math>\nabla:A\otimes A\rightarrow A<math>of the degree of the identity of G.

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

See also

es:lgebra graduada ru:Градуированная алгебра

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