Graded algebra
From Academic Kids
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:
- 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
- <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).
