- Graded algebra
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 gradation (or "grading").
A graded ring "A" is a ring that has a
direct sumdecomposition into (abelian) additive groups:such that the ring multiplication maps:Explicitly this means that:and so:
Elements of are known as "homogeneous elements" of degree "n". An ideal or other subset ⊂ "A" is homogeneous if for every element "a" ∈ , the homogeneous parts of "a" are also contained in
If "I" is a homogeneous ideal in "A", then is also a graded ring, and has decomposition:.
Any (non-graded) ring "A" can be given a gradation by letting "A"0 = "A", and "A""i" = 0 for "i" > 0. This is called the trivial gradation on "A".
The corresponding idea in
module theoryis that of a graded module, namely a module "M" over a graded ring "A" such that also
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).
A graded algebra over a graded ring "A" is an "A"-algebra "E" which is both a graded "A"-module and a graded ring in its own right. Thus "E" admits a direct sum decomposition::such that
#"A"i"E"j ⊂ "E"i+j, and
#"E"i"E"j ⊂ "E"i+j.
Often when no grading on "A" is specified, it is assumed that "A" receives the trivial gradation, in which case one may still talk about graded algebras over "A" without risk of confusion.
Examples of graded algebras are common in mathematics:
Polynomial rings. The homogeneous elements of degree "n" are exactly the homogeneous polynomials of degree "n".
tensor algebra"T"•"V" of a vector space"V". The homogeneous elements of degree "n" are the tensors of rank "n", "T""n""V".
exterior algebraΛ•"V" and symmetric algebra"S"•"V" are also graded algebras.
cohomology ring"H"• in any cohomology theoryis also graded, being the direct sum of the "H""n".
Graded algebras are much used in
commutative algebraand algebraic geometry, homological algebraand algebraic topology. One example is the close relationship between homogeneous polynomials and projective varieties.
G-graded rings and algebras
We can generalize the definition of a graded ring using any
monoid"G" as an index set. A "G"-graded ring "A" is a ring with a direct sum decomposition:such that:
*A graded algebra is then the same thing as a N-graded algebra, where N is the monoid of non-negative integers.
*If we do not require that the ring have an identity element,
semigroups may replace monoids.
*"G"-graded modules and algebras are defined in the same fashion as above.
* A group naturally grades the corresponding
group ring; similarly, monoid rings are graded by the corresponding monoid.
superalgebrais another term for a Z2-graded algebra. Examples include Clifford algebras. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).
category theory, a "G"-graded algebra "A" is an object in the category of "G"-graded vector spaces, together with a morphism of the degree of the identity of "G".
Some graded rings (or algebras) are endowed with an
anticommutativestructure. This notion requires the use of a semiringto supply the gradation rather than a monoid. Specifically, a signed semiring consists of a pair (Γ, ε) where Γ is a semiringand ε : Γ → Z/2Z is a homomorphismof additive monoids. An anticommutative Γ-graded ring is a ring "A" graded with respect to the "additive" structure on Γ such that::xy=(-1)ε (deg x) ε (deg y)yx, for all homogeneous elements "x" and "y".
exterior algebrais an example of an anticommutative algebra, graded with respect to the structure (Z≥ 0, ε) where ε is the homomorphism given by ε("even") = 0, ε("odd") = 1.
supercommutative algebra(sometimes called a skew-commutative associative ring) is the same thing as an anticommutative (Z/2Z, ε) -graded algebra, where ε is the identity endomorphism for the additive structure.
graded vector space
differential graded algebra
graded Lie algebra
filtered algebra, a generalization
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