Sunday, July 09, 2006

Differential graded Hopf algebras in Factor

The number of people interested in both Algebraic Topology and Factor is probably exactly zero, however here goes anyway. If you are interested in both topics, please let me know, we have a lot to discuss.

I implemented a library for working with finitely-generated connected Hopf algebras over the complex numbers. The code can be found in the darcs repository. While Hopf algebras can be quite abstract, finitely-generated connected Hopf algebras can be realized as a tensor product of a polynomial algebra together with a exterior algebra.

Elements are added using h+, multiplied using /\ and printed using h.. The unit is 1 and the co-unit is co1. I have not implemented co-multiplication yet, since I'm not sure how to represent an element of the tensor product of two Hopf algebras.

One starts by defining some generators, and their degrees:
  SYMBOLS: a b c u ;
1 a deg=
1 b deg=
1 c deg=
2 u deg=

Every Hopf algebra embeds a copy of the ground field:
  3 4 h+ h.
10 2 /\ h.

Even degree generators behave like polynomial indeterminites:
  u u /\ 3 /\ u 4 /\ h+ h.
4u + 3u/\u

Odd degree elements "anti-commute:"
  b a /\ h.
- a/\b

You can also define a differential -- this is of most interest to me personally, since my masters thesis will most likely focus on cohomology of Lie algebras.

Suppose we want the derivative of u to be abc:
  a b /\ c /\ u d=

The derivative of a product is evaluated using the Liebnitz rule (d(u/\v)=du/\v + (-1)^deg(v) u/\dv).

Now we can differentiate some expressions:
  u u /\ d h.

An expression whose derivative is zero is known as a cycle:
  u a /\ d h.

An element in the range of the differential is known as a boundary. It is a theorem that all boundaries are cycles; that is, applying d twice yields zero:
  u u /\ d d h.

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