Quantization of Lie bialgebras and shuffle algebras of Lie algebras
| dc.creator | Enriquez, B. | |
| dc.date | 2000-08-16 | |
| dc.date | 2000-09-28 | |
| dc.date.accessioned | 2026-07-07T04:36:51Z | |
| dc.date.available | 2026-07-07T04:36:51Z | |
| dc.description | To any field K of characteristic 0, we associate a set Sha(K). Elements of Sha(K) are equivalence classes of families of Lie polynomials subject to associativity relations. We construct an injection and a retraction between Sha(K) and the set of quantization functors of Lie bialgebras over K. This construction involves the following steps. 1) To each element \varpi of Sha(K), we associate a functor g\mapsto Sh(g) from the category of Lie algebras to that of Hopf algebras; Sh(g) contains Ug. 2) When g and h are Lie algebras, and r_{gh} \in g\otimes h, we construct an element R(r_{gh}) of Sh(g)\otimes Sh(h) satisfying quasitriangularity identities; R(r_{gh}) defines a Hopf algebra morphism from Sh(g)^* to Sh(h). 3) When g = h and r\in g\otimes g is a solution of CYBE, we construct a series ρ(r) such that R(ρ(r)) is a solution of QYBE. The expression of ρ(r) in terms of r involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing statements for cohomologies arising from universal algebras for the solutions of CYBE. 4) We define the quantization of a Lie bialgebra g as the image of the morphism defined by R(ρ(r)), where r\in g\otimes g^* is the canonical element attached to g. | |
| dc.identifier | https://arxiv.org/abs/math/0008128 | |
| dc.identifier | http://arxiv.org/abs/math/0008128 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/59748 | |
| dc.subject | Quantum Algebra | |
| dc.title | Quantization of Lie bialgebras and shuffle algebras of Lie algebras | |
| dc.type | text |