Quantization of Lie bialgebras and shuffle algebras of Lie algebras

Details Quantization of Lie bialgebras and shuffle algebras of Lie algebras Quantization of Lie bialgebras and shuffle algebras of Lie algebras

2000-08-16

arxiv.org/abs/math/0008128v6

Mathematics

Quantum Algebra

Scientific paper

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 \rho(r) such that R(\rho(r)) is a solution of QYBE. The expression of \rho(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(\rho(r)), where r\in g\otimes g^* is the canonical element attached to g.

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