Groups and Lie algebras corresponding to the Yang–Baxter equations
By Laurent Bartholdi, B. Enriquez, Pavel Etingof, Eric M. Rains. 2005
50 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Geometry and Topology, Molecular Biology
View Paper: https://www.semanticscholar.org/paper/5816f2ddd5863dfee1ac6ac77ee5fbc796e812a9
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In Algebra and Number Theory
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Citations in the corpus
- FI_W-modules and stability criteria for representations of the classical Weyl groups (2013, background): “For details see for example Kauffman [Kau99, Kau00], Vershinin [Ver01], Kauffman– Lambropoulou [KL04], Bardakov [Bar04], and Bartholdi–Enriquez–Etingof– Rains [BEER06].”
- K(π, 1) and word problems for infinite type Artin–Tits groups, and applications to virtual braid groups (2012, background): “Remark A classifying space QCn for the pure virtual braid group V Pn is constructed in [9].”