Groups and Lie algebras corresponding to the Yang–Baxter equations
| Autor: | Laurent Bartholdi, Pavel Etingof, Eric M. Rains, Benjamin Enriquez |
|---|---|
| Přispěvatelé: | Institut de Recherche Mathématique Avancée (IRMA), Université Louis Pasteur - Strasbourg I-Centre National de la Recherche Scientifique (CNRS) |
| Rok vydání: | 2006 |
| Předmět: |
Pure mathematics
Non-associative algebra Universal enveloping algebra 0102 computer and information sciences Group Theory (math.GR) 01 natural sciences Representation theory [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] Physical Sciences and Mathematics FOS: Mathematics math.GR 0101 mathematics math.RA Mathematics Discrete mathematics Algebra and Number Theory Simple Lie group 010102 general mathematics [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA] Mathematics - Rings and Algebras Killing form Affine Lie algebra Lie conformal algebra Adjoint representation of a Lie algebra 010201 computation theory & mathematics Rings and Algebras (math.RA) Mathematics - Group Theory |
| Zdroj: | Journal of Algebra Journal of Algebra, Elsevier, 2006, 305 (2), pp.742-764 Journal of Algebra, Elsevier, 2006, 305 (2), pp.742-764. ⟨10.1016/j.jalgebra.2005.12.006⟩ Bartholdi, Laurent; Enriquez, Benjamin; Etingof, Pavel; & Rains, Eric. (2005). Groups and Lie algebras corresponding to the Yang-Baxter equations. J. Algebra 305 (2006), no. 2, 742--764. doi: 10.1016/j.jalgebra.2005.12.006. UC Davis: Department of Mathematics. Retrieved from: http://www.escholarship.org/uc/item/58d7w70j |
| ISSN: | 0021-8693 1090-266X |
| DOI: | 10.1016/j.jalgebra.2005.12.006 |
| Popis: | For a positive integer n we introduce quadratic Lie algebras tr_n qtr_n and 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 find their Hilbert series. We also compute the cohomology rings of these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). 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. We conjecture that this map is actually an isomorphism (this is now a theorem due to P. Lee). At the same time, we show that the groups Tr_n and QTr_n are not formal for n>3. Errors pointed out by P. Lee corrected. Proposition 5.1 in the previous version was incorrect, so Theorems 2.3 and 8.5 and Proposition 6.1 were proved incorrectly. In the current version, Propositions 5.1 and 6.1 are deleted and Theorems 2.3, 8.5 are stated as conjectures. They were proved by Peter Lee, along with Conjectures 2.4 and 8.6 of the previous version |
| Databáze: | OpenAIRE |
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