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Monograph, 73 pages : PACIFIC JOURNAL OF MATHEMATICS, Vol. 195, No. 2, 2000

Abstract:

The authors construct for any finite-dimensional Lie bialgebra [g], a bialgebra A_u,v([g]) over the ring C[u][[v]; then they prove that this bialgebra quantizes simultaneously the universal enveloping bialgebra U([g]), the bialgebra dual to U([g]), and the symmetric bialgebra S([g]). As previously proposed by Turaev, Au,v([g]) is a biquantization of S([g]). It is then claimed that: " Au,v([g]) contains all information about the quantization of [g]". A verbatim quote from the authors introduction: "The notion of a Lie bialgebra was introduced by Drinfeld (1982, 1987) in the framework of his algebraic formalism for the quantum inverse scattering method. A Lie bialgebra is a Lie algebra [g] provided with a Lie cobracket g --> g (oX) g which is related to the Lie bracket by a certain compatibility condition. The notion of a Lie bialgebra is self-dual: If [g ]is a finite-dimensional Lie bialgebra over a field, then the dual g* is also a Lie bialgebra." Free download at: http://nyjm.albany.edu:8000/PacJ/p/2000/195-2-3.pdf http://nyjm.albany.edu:8000/PacJ/2000/195-2-3.html

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