Grading and Filtration

Graded Algebra and Filtered Algebra

Graded Algebra

A graded algebra is an algebra with a linear decomposition such that

• each is a sub vector space of ,
• , implies ,

We call all homogeneous (elements) of degree . Suppose and are graded algebras, then a map between graded algebras is a graded homomorphism if . Hence we have a category of graded algebras .

Degree Completed Graded Algebra

A degree completed graded algebra is a degree completion of a graded algebra by taking the inverse limit.

Direct sum vs. Direct product

As vector spaces, the only difference is that the former consists of finite linear combinations of homogeneous elements, whilst in the latter, we also have infinite linear combinations: think power series versus polynomials.

Warning

We will allow a graded algebra to be either a direct sum, or a direct product, and will restrict to graded algebras which are direct products (degree completed graded algebra), and just call it graded algebra, as this is the most common case in knot theory.

e.g. Suppose is polynomials in variables, then

Remark

Graded algebra is good in the sense that:

• degree is a notion of complexity (of computation).
• easy to deal with computation (degree by degree or up to degree )
• can apply induction on degree

Filtered Algebra

A filtered algebra is an algebra with a sequence of nested ideals: such that , implies . Suppose and are filtered algebras, then an algebra homomorphism between filtered algebras is a filtered homomorphism if . Hence we have a category of filtered algebras .

Remark

This is not as good as graded algebra because, for instance, though , implies , as , could actually be in .

e.g. Any graded algebra is a filtered algebra, since we can take . And is an ideal because any and implies .

The Associated Graded Functor

Associated Graded Algebra

Given a filtered algebra , we can define a graded algebra by taking . We denote this graded algebra as , and define the multiplication of components by and extend to all elements of via Cauchy product formula: for all .

Proposition

forms a functor from to .

Proof Suppose and are filtered algebras, and is a filtered homomorphism. Then we have a graded homomorphism defined by It is clear that is graded, because . Moreover, is a functor because it preserves composition and identity.

The Pro-Unipotent Filtration

In this section we define a famous filtration on the group algebra , called the pro-unipotent filtration, studied extensively in group theory and algebraic geometry.

Augmentation Map

The augmentation map is defined by sending each group element to : where and .

Proof The augmentation map is clearly an algebra homomorphism because .

Augmentation Ideal

An augmentation ideal is the kernel of the augmentation map. That is, the set of elements of whose coefficients sum to zero.

Proposition

The product of augmentation ideal for times is an ideal of the algebra. i.e. .

Proof For all , we have .

Pro-unipotent Filtration

The pro-unipotent (a.k.a -adic) filtration of a an algebra is the sequence of ideals: where is the product of augmentation ideal for times, that is the ideal generated by products of elements of .

e.g. Consider , which is the set of Laurent polynomials, where we identify by convention. Then the augmentation ideal is the principle ideal generated by . Because we can write any element in as which is a linear combination of for . Hence as an ideal, is generated by . And thus one can easily show that . And it turns out that is the ring of formal power series in .