GK dimension of the relatively free algebra for $$sl_2$$ s l 2

Abstract

Let \(sl_2(K)\) be the Lie algebra of the \(2\times 2\) traceless matrices over an infinite field \(K\) of characteristic different from 2, denote by \(R_m= R_m(sl_2(K))\) the relatively free (also called universal) algebra of rank \(m\) in the variety of Lie algebras generated by \(sl_2(K).\) In this paper we compute the Gelfand–Kirillov dimension of the Lie algebra \(R_m(sl_2(K)).\) It turns out that whenever \(m\ge 2\) one has \(\mathrm{GK}\dim R_m = 3(m-1).\) In order to compute it we use the explicit form of the Hilbert series of \(R_m\) described by Drensky. This result is new for \(m>2\); the case \(m=2\) was dealt with by Bahturin in 1979.