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.

7

10 2 /\ h.

20

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.

2a/\b/\c/\u

An expression whose derivative is zero is known as a

*cycle*:

u a /\ d h.

0

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.

0

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