NationMaster - Encyclopedia: Graded algebra

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). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

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

Graded rings

A graded ring A is a ring that has a direct sum decomposition into (abelian) additive groups In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

$A = bigoplus_{nin mathbb N}A_i = A_0 oplus A_1 oplus A_2 oplus cdots$

such that the ring multiplication maps

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

Explicitly this means that whenever

$s in A_s, r in A_r implies sr in A_{s+r}$

and so

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

Elements of $A n$ are known as homogeneous elements of degree n. 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 $mathfrak{a}.$ 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 In abstract algebra, a module is a generalization of a vector space. ... 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 to lie in a field, the scalars may lie in an arbitrary ring. ...

$M = bigoplus_{iin mathbb N}M_i ,$

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). In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... In abstract algebra, the length of a module is a measure of the modules size. It is defined as the length of the longest ascending chain of submodules and is a generalization of the concept of dimension for vector spaces. ... In mathematics, the Hilbert polynomial of a graded commutative algebra A = âŠ•An over a field k that is generated by the finite dimensional space A1 is the unique polynomial f(x) with rational coefficients such that f(n) = dimk An for all but finitely many positive integers n. ...

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. 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. In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, the tensor algebra of a vector space V, denoted T(V) or Tâ€¢(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann[1]) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... This article is about algebraic 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 In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...

$A = bigoplus_{iin G}A_i$

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.
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.

Examples: In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...

A group naturally grades the corresponding group ring; similarly, monoid rings are graded by the corresponding monoid.
A superalgebra is another term for a Z₂-graded algebra. Examples include Clifford algebras. 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 $nabla:Aotimes Arightarrow A$ of the degree of the identity of G. In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described... In abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid. ... In mathematics and theoretical physics, a superalgebra over a field K generally refers to a Z2-graded algebra over K (here Z2 is the cyclic group of order 2). ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... Clifford algebras are a type of associative algebra in mathematics. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, Vect(K, Z2) denotes the category whose objects are all of the Z2-graded vector spaces over the given field K. The morphisms of this category are given by the even and odd linear transformations between any two such objects. ...

Anticommutativity

Some graded rings (or algebras) are endowed with an anticommutative structure. This notion requires the use of a semiring to supply the gradation rather than a monoid. 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. An anticommutative Γ-graded ring is a ring A graded with respect to the additive structure on Γ such that: A mathematical operator (typically a binary operator, represented by *) is anticommutative iff it is true that x * y = −(y * x) for all x and y on the operators valid domain (e. ... In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ... In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ... In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ...

xy=(-1)^{ε (deg x) ε (deg y)}yx, for all homogeneous elements x and y.

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. ...

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, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann[1]) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... In mathematics, a supercommutative algebra is a superalgebra (i. ...

Contents

Graded modules

Graded algebras

G-graded rings and algebras

Anticommutativity

Examples

See also