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Groups and Lie algebras corresponding to the Yang–Baxter equations

Bartholdi, Laurent and Enriquez, Benjamin and Etingof, Pavel and Rains, Eric (2006) Groups and Lie algebras corresponding to the Yang–Baxter equations. Journal of Algebra, 305 (2). pp. 742-764. ISSN 0021-8693. doi:10.1016/j.jalgebra.2005.12.006.

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For a positive integer n we introduce quadratic Lie algebras tr_n, qtr_n and finitely discrete groups Tr_n, QTr_n naturally associated with the classical and quantum Yang–Baxter equation, respectively. We prove that the universal enveloping algebras of the Lie algebras tr_n, qtr_n are Koszul, and compute their Hilbert series. We also compute the cohomology rings for these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). Finally, we construct a basis of U(tr_n). We construct cell complexes which are classifying spaces of the groups Tr_n and QTr_n, and show that the boundary maps in them are zero, which allows us to compute the integral cohomology of these groups. We show that the Lie algebras tr_n, qtr_n map onto the associated graded algebras of the Malcev Lie algebras of the groups Tr_n, QTr_n, respectively. In the case of Tr_n, we use quantization theory of Lie bialgebras to show that this map is actually an isomorphism. At the same time, we show that the groups Tr_n and QTr_n are not formal for n⩾4.

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Additional Information:© 2006 Elsevier Inc. Received 28 September 2005, Available online 17 February 2006. The authors are very grateful to: Richard Stanley for help with the statement and proof of Proposition 5.1—this allowed us to significantly strengthen the main results of the paper; André Henriques for contributing a proof of Theorem 8.1; and Leonid Bokut for pointing out an error in the previous version and for giving useful references. P.E. and B.E. thank the Mathematics Department of ETH (Zürich) for hospitality. The work of P.E. was partially supported by the NSF grant DMS-0504847 and the CRDF grant RM1-2545-MO-03. E.R. was supported in part by NSF Grant No. DMS-0401387. Throughout the work, we used the “Magma” program for algebraic computations [Ma].
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Civilian Research & Development Foundation (CRDF)RM1-2545-MO-03
Issue or Number:2
Record Number:CaltechAUTHORS:20171002-155152258
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Official Citation:Laurent Bartholdi, Benjamin Enriquez, Pavel Etingof, Eric Rains, Groups and Lie algebras corresponding to the Yang–Baxter equations, Journal of Algebra, Volume 305, Issue 2, 15 November 2006, Pages 742-764, ISSN 0021-8693, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81974
Deposited By: Tony Diaz
Deposited On:02 Oct 2017 23:08
Last Modified:15 Nov 2021 19:47

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