mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a filtered algebra is a generalization of the notion of a

graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...

. Examples appear in many branches of

, especially in

homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

and

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

. A filtered algebra over the field

k

is an

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

(A,\cdot)

over

k

that has an increasing sequence

\ \subseteq F_0 \subseteq F_1 \subseteq \cdots \subseteq F_i \subseteq \cdots \subseteq A

of subspaces of

A

such that :

A=\bigcup_ F_

and that is compatible with the multiplication in the following sense: :

\forall m,n \in \mathbb,\quad F_m\cdot F_n\subseteq F_.

Associated graded algebra

In general, there is the following construction that produces a graded algebra out of a filtered algebra. If

A

is a filtered algebra, then the '' associated graded algebra''

\mathcal(A)

is defined as follows: The multiplication is well-defined and endows

\mathcal(A)

with the structure of a graded algebra, with gradation

\_.

Furthermore if

A

associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

then so is

\mathcal(A)

. Also, if

A

is unital, such that the unit lies in

F_0

, then

\mathcal(A)

will be unital as well. As algebras

A

and

\mathcal(A)

are distinct (with the exception of the trivial case that

A

is graded) but as vector spaces they are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

. (One can prove by induction that

\bigoplus_^nG_i

is isomorphic to

F_n

as vector spaces).

Examples

Any

graded by

\mathbb

, for example

A = \bigoplus_ A_n

, has a filtration given by

F_n = \bigoplus_^n A_i

. An example of a filtered algebra is the

Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...

\operatorname(V,q)

of a vector space

V

endowed with a

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

q.

The associated graded algebra is

\bigwedge V

, the

exterior algebra In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...

V.

The symmetric algebra on the dual of an

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

is a filtered algebra of

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s; on a

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

, one instead obtains a graded algebra. The universal enveloping algebra of a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

\mathfrak

is also naturally filtered. The PBW theorem states that the associated graded algebra is simply

\mathrm (\mathfrak)

. Scalar

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

s on a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

M

form a filtered algebra where the filtration is given by the degree of differential operators. The associated graded algebra is the

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

of smooth functions on the

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...

T^*M

which are polynomial along the fibers of the projection

\pi\colon T^*M\rightarrow M

. The group algebra of a group with a length function is a filtered algebra.

References

* {{PlanetMath attribution, id=3938, title=Filtered algebra Algebras Homological algebra

Associated graded algebra

Examples

See also

References