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Quantization of Lie bialgebras. I. (English) Zbl 0863.17008
Let $$L$$ be a Lie bialgebra over a field $$k$$ of characteristic zero. The authors construct a quantization of $$L$$ in the sense of Drinfeld, thus answering a question of V. G. Drinfeld [Lect. Notes Math. 1510, 1–8 (1992; Zbl 0765.17014)]. In the first part of the paper, the construction is given for $$L$$ finite-dimensional, $$G= L\oplus L^*$$ its double. $$M$$ is the category of $$G$$-modules, which is given the structure of a braided monoidal category. Verma modules $$M_+$$ and $$M_-$$ over $$G$$ are used to construct a fiber functor from $$M$$ to the tensor category of topologically free $$k[[h]]$$ modules, which yields a (topological) Hopf algebra $$H$$ isomorphic to $$U(G)[[h]]$$. The quantization of $$L$$ is a certain Hopf subalgebra $$H_+$$ of $$H$$, whose quantum double is $$H$$. As an application, every classical $$r$$-matrix over an associative $$k$$-algebra $$A$$ can be quantized, i.e., there is a quantum $$R$$-matrix $$R$$ in $$(A\otimes A) [[h]]$$ such that $$R=1+ hr+O(h^2)$$. The construction in part I is neither functorial nor universal.
In part II, a general construction is given for any Lie algebra $$L$$ (possibly infinite-dimensional). The double $$G$$ of $$L$$ will have a nontrivial topology if $$L$$ is infinite-dimensional. One considers the category $$M^e$$ of equicontinuous $$G$$-modules, which is given a braided monoidal structure. Verma modules $$M_+$$ and $$M_-$$ over $$G$$ are constructed. $$M_-$$ is equicontinuous, as is the dual module $$M^*_+$$. A fiber function $$F$$ from $$M^e$$ to the category of topological $$k[[h]]$$-modules is defined. $$F$$ is not necessarily representable, but has a tensor structure which agrees with the one in part I when $$L$$ is finite-dimensional. Let $$H= \text{End} F$$. The quantization of $$L$$ is a certain subalgebra $$H_+$$ of $$H$$ and $$\Delta H_+ \subset H_+ \otimes H_+$$. If $$L$$ is finite-dimensional, $$H_+$$ coincides with the one in Part I. The construction is both functorial and universal, which will be explained in a subsequent paper.

MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B62 Lie bialgebras; Lie coalgebras 16T05 Hopf algebras and their applications 18M15 Braided monoidal categories and ribbon categories
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References:
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