graded algebra

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.

Graded algebra

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

A = \bigoplus_{n\in N}A_i

such that

A_m A_n \subseteq A_{m + n}.

A graded algebra is a special case of a graded vector space. Elements of

A_n

are known as homogeneous elements of degree n. Examples of graded algebras are common in mathematics:

Polynomial rings. The homogeneous elements of degree n are exactly the homogeneous polynomials of degree n.
The tensor algebra T^•V and exterior algebra Λ^•V of a vector space V. The homogeneous elements of degree n are T^•V and Λ^•V, respectively.
The cohomology ring H^• in any cohomology theory is also graded by N, being the direct sum of the Hⁿ.

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

A = \bigoplus_{i\in G}A_i

such that

A_i A_j \subseteq A_{i \cdot j}

An element of the ith subspace A_i 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.
Any discrete valuation ring is N-graded by powers of the maximal ideal; further still, at least when the elements in the monoid G lack inverses, the monoid ring RG will be G-graded.

Category theoretically, a G-graded algebra A is an object in the category of G-graded vector spaces together with a morphism

\nabla:A\otimes A\rightarrow A

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).

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Graded Algebra

Graded algebra

G-graded algebra

See also