There are lots of papers dealing with representation-theoretic questions and universal enveloping algebras using Gr&ouml;bner bases. Some examples are given by these: <a href="http://s-space.snu.ac.kr/bitstream/10371/12179/1/42.pdf">1</a>, <a href="http://www.ams.org/journals/spmj/2011-22-04/S1061-0022-2011-01159-8/home.html">2</a>, <a href="http://www.springerlink.com/content/bwn7336156456u7u/">3</a>, <a href="http://vm-jn.wspc.com.sg/ijac/06/0604/S0218196796000234.html">4</a>.