Gröbner Bases
Syzygies
Resolutions
Quantum Alg.
Left Maximality
Max. Twosided
Finding a Maximal Two-Sided Ideal in A Given Left Ideal
This application is particularly important for representation theory.
Suppose we have given a left ideal
L0 = L.
We are computing the descending sequence of left ideals
Lk+1 := { a in Lk | axi in Lk for all i},
where x1 ,..., xn are the algebra variables.

Then the maximal two-sided ideal in L is the intersection of all Lk, so we can obtain its generators as invariants of the transitions Lk --> Lk+1.

We demonstrate the computations for the algebra

U(sl2) = < e, f, h | fe = ef - h, he = eh + 2e, hf = fh - 2f >
and the ideal L = <e, h-a> , where a is considered as a parameter.

PLURAL computations
KL, 06/03 http://www.singular.uni-kl.de