Graded algebra

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

1 Graded algebra
2 G-graded algebra
3 Graded modules
4 See also

Graded algebra

A graded algebra A is an algebra that has a direct sum decomposition

$A = \bigoplus_{n\in \mathbb N}A_i = A_0 \oplus A_1 \oplus A_2 \oplus \cdots$

such that

$A_m A_n \subseteq A_{m + n}.$

Elements of $A n$ are known as homogeneous elements of degree n. An ideal, or other set in A, is homogeneous if for every element a it contains, the homogeneous parts of a are also contained in it.

Since rings may be regarded as Z-algebras, a graded ring is defined to be a graded Z-algebra.

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 of a vector space V. The homogeneous elements of degree n are the tensors of rank n, TⁿV.
The exterior algebra Λ^•V and symmetric algebra S^{•^{V are also graded algebras.}}
The cohomology ring H^• in any cohomology theory is also graded, 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 G as an index set. A G-graded algebra A is an algebra with a direct sum decomposition

$A = \bigoplus_{i\in G}A_i$

such that

$A_i A_j \subseteq A_{i \cdot j}$

A graded algebra is then the same thing as a N-graded algebra, where N is the monoid of natural numbers.

(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.
A superalgebra is another term for a Z₂-graded algebra. Clifford algebras are a common family of examples. Here the homogeneous elements are either even (degree 0) or odd (degree 1).

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.

Graded modules

The corresponding idea in module theory is that of a graded module, namely a module M over A such that also

$M = \bigoplus_{n\in \mathbb N}M_i ,$

and

$A_iM_j \subseteq M_{i+j}$

This idea is much used in commutative algebra, and elsewhere, to define under mild hypotheses a Hilbert function, namely the length of M_n as a function of n. Again under mild hypotheses of finiteness, this function is a polynomial, the Hilbert polynomial, for all large enough values of n (see also Hilbert-Samuel polynomial).

Graded algebra

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Graded algebra

G-graded algebra

Graded modules

See also

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