[GAP Forum] Computation using GAP

[yilan tan]

Dear All:

I am a Ph.D student at the University of Alberta. My research interest is
the representation theory of Yangian. I used GAP to find weight of weight
vectors, quickly and efficiently. I feel GAP is so powerful. I would like
to seek more applications of GAP to my research. Here is one of my
questions.

Suppose that I have some elements from an algebra, $h_i$ and $x_i$, where
$0\leq i\leq 6$. The defining relations are: 1. $[h_0,x_i]=cx_i$, where c
is a natural number and [, ] denoting the Lie bracket.
2.$[h_i,x_j]=[h_{i-1},x_{j+1}]+(c/2)(h_{i-1}x_{j}+x_jh_{i-1})$.

We can get at least two things immediately from the defining relations.
3. $[h_i,x_0]=\sum_{s} c_s x_s+\sum_{a,b} c_{a, b} x_ah_b$, here both $c_s$
and $c_ {a, b}$ are rational numbers.
4.inductively, $[h_i,(x_0)^m]=\sum_{s_1,\ldots,
s_m}c_{s_1,\ldots,s_m)x_{s_1}\ldots x_{s_m)+\sum_{a_1,\ldots, a_m,b}
c_{a_1,\ldots, a_m,b} x_{s_1}\ldots x_{s_m)h_b$.

If some of you can kindly answer my questions:

1. How to define the orders of x_j precedes h_i. Which chapters I should
read?

2. How to define (3) in GAP system?

3. If it is possible to define an associative algebra with infinite
generators?

Thank you very much for reading my question. I would appreciate if you can
comment my questions.

All the best.

Peter Tan