2019/05/18 by Natalia K. Iyudu, Iyudu, Natalia K., Susan J. Sierra +1 · 3 citations
Mathematics · Physics and Astronomy · #16P90 #16S30 #17B65 #17B68 #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math-ph #math.AG #math.MP #math.QA #math.RA #math.RT #msc:16P90 #msc:16S30 #msc:17B65 #msc:17B68
paper · pdf · doi:10.48550/arxiv.1905.07507
Version accepted for publication in Arkiv for Matematik
arxiv created 2020/03/04 · arxiv updated 2020/03/05
Let \mf g be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra U(\mf g ) has intermediate growth and thus infinite Gelfand-Kirillov (GK-) dimension. We prove that the GK-dimension of U(\mf g) is \em just infinite in the sense that any proper quotient of U(\mf g) has polynomial growth. This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. We also establish the corresponding results for quotients of the symmetric algebra S(\mf g) by proper Poisson ideals. In fact, we prove more generally that any central quotient of the universal enveloping algebra of the Virasoro algebra has just infinite GK-dimension. We give several applications. In particular, we easily compute the annihilators of Verma modules over the Virasoro algebra.