On Quantum Spaces of Lie Algebras

Abstract

The homogenization H(g) of the enveloping algebra of a finite dimensional Lie algebra 0 is an Artin-Schelter regular algebra.We characterize rf-dimensional linear subspaces in the corresponding quantum space P?(0) as homogenizations of induced representations from codimension d Lie subalgebras.Furthermore we prove that the point variety has an embedded component iff there is a line, not contained in this point variety. Introduction and notationTo any finite-dimensional Lie algebrak=i one classically associates an associative algebra U(g), the so-called "enveloping algebra" of g.The enveloping algebra carries a natural filtration Uo(q)cUx(q)c---cU"(q)c--such that gr U(g>) = S(q) , the symmetric algebra on a..This is the famous Poincare-Birkhoff-Witt theorem.By homogenizing the defining relations of U(q) one can also associate a regular algebra in the sense of Artin and Schelter [1] to g.This algebra, which will be denoted by H(q) in the sequel, is the positively graded quadratic algebra generated by Xq, ... , Xn where Xq is taken to be central and the remaining defining relations are n XiXj -XjXj -2J o.ijkXkXQ .i=l Observe that H(q)/(X0 -I)H(q) = U(g), the enveloping algebra of q , and that H(q)/XqH(q) = gr U(q)) = S(q) .From these facts one derives as usual (see, e.g., [8] or [9]) that H(g) is a Noetherian maximal order domain, that it is Auslander-regular of dimension n + 1, and that it satisfies the Cohen-Macaulay property.