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.
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Thanks are due to the Referees who appointed a gap in the first version of the paper, and whose remarks improved significantly the exposition.
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Communicated by C. Krattenthaler.
Supported by Ph.D. Grant from CAPES and from CNPq.
P. Koshlukov partially supported by grants from CNPq (No. 304003/2011-5 and 480139/2012-1), and from FAPESP (No. 2010/50347-9).
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Machado, G.G., Koshlukov, P. GK dimension of the relatively free algebra for \(sl_2\) . Monatsh Math 175, 543–553 (2014). https://doi.org/10.1007/s00605-014-0687-2
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DOI: https://doi.org/10.1007/s00605-014-0687-2