Graded algebra - Citizendia

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 gradation (or grading). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property

1 Graded rings
2 Graded modules
3 Graded algebras
4 G-graded rings and algebras
5 Anticommutativity
- 5.1 Examples
6 See also
7 References

Graded rings

A graded ring A is a ring that has a direct sum decomposition into (abelian) additive groups

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

such that the ring multiplication maps

$A_s \times A_r \rightarrow A_{s + r}.$

Explicitly this means that

$x \in A_s, y \in A_r \implies xy \in A_{s+r}$

and so

$A_s A_r \subseteq A_{s + r}.$

Elements of $A n$ are known as homogeneous elements of degree n. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction An ideal or other subset $\mathfrak{a}$ ⊂ A is homogeneous if for every element a ∈ $\mathfrak{a}$ , the homogeneous parts of a are also contained in $\mathfrak{a}.$

If I is a homogeneous ideal in A, then $A / I$ is also a graded ring, and has decomposition

$A/I = \bigoplus_{n\in \mathbb N}(A_n + I)/I$ . In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

Any (non-graded) ring A can be given a gradation by letting A₀ = A, and A_i = 0 for i > 0. This is called the trivial gradation on A.

Graded modules

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

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

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. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure In Abstract algebra, the length of a module is a measure of the module's "size" 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). In Commutative algebra, the Hilbert polynomial of a graded commutative algebra or Graded module is a Polynomial in one variable that measures

Graded algebras

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

$E=\bigoplus_i E_i$

such that

A_iE_j ⊂ E_i+j, and
E_iE_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. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables The homogeneous elements of degree n are exactly the homogeneous polynomials of degree n. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations
The tensor algebra T^•V of a vector space V. In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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. In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field
The cohomology ring H^• in any cohomology theory is also graded, being the direct sum of the Hⁿ. In Mathematics, specifically Algebraic topology, the cohomology ring of a Topological space X is a ring formed from the Cohomology In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain

Graded algebras are much used in commutative algebra and algebraic geometry, homological algebra and algebraic topology. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic One example is the close relationship between homogeneous polynomials and projective varieties. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety

G-graded rings and algebras

We can generalize the definition of a graded ring using any monoid G as an index set. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation A G-graded ring A is a ring with a direct sum decomposition

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

such that

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

Remarks:

A graded algebra is then the same thing as a N-graded algebra, where N is the monoid of non-negative integers. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an
If we do not require that the ring have an identity element, semigroups may replace monoids. In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation
G-graded modules and algebras are defined in the same fashion as above.

Examples:

A group naturally grades the corresponding group ring; similarly, monoid rings are graded by the corresponding monoid. In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively In Abstract algebra, a monoid ring is a new ring constructed from some other ring and a Monoid.
A superalgebra is another term for a Z₂-graded algebra. In Mathematics and Theoretical physics, a superalgebra is a Z 2- Graded algebra. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Examples include Clifford algebras. In Mathematics, Clifford algebras are a type of Associative algebra. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).

In category theory, 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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a graded vector space is a type of Vector space that includes the extra structure of gradation, meaning that it can be composed into the

Anticommutativity

Some graded rings (or algebras) are endowed with an anticommutative structure. In mathematics anticommutativity refers to the property of an operation being anticommutative, i This notion requires the use of a semiring to supply the gradation rather than a monoid. In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse Specifically, a signed semiring consists of a pair (Γ, ε) where Γ is a semiring and ε : Γ → Z/2Z is a homomorphism of additive monoids. In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector 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.

Examples

An exterior algebra is 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.
A 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. In Mathematics, a supercommutative algebra is a Superalgebra (i

References

Lang, Serge (2002), Algebra, vol. In Mathematics, a graded vector space is a type of Vector space that includes the extra structure of gradation, meaning that it can be composed into the A graded category is a mathematical concept If \mathcal{A} is a category, then a \mathcal{A}-graded category is a category In Mathematics, in particular Abstract algebra and Topology, a differential graded algebra is a Graded algebra with an added chain complex structure In Mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In Mathematics, a filtered algebra is a generalization of the notion of a Graded algebra. Serge Lang ( May 19, 1927 – September 12, 2005) was a French -born American Mathematician. 211 (3rd ed. ), Graduate Texts in Mathematics, Springer-Verlag .

Contents