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.

Graded algebra

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

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such that

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Elements of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

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

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such that

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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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

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and

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This idea is much used in commutative algebra, and elsewhere, to define under mild hypotheses a Hilbert function, namely the length of Mn 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).

See also

de:Graduierung (Algebra) es:Álgebra graduada ru:Градуированная алгебра