Quantization of Lie bialgebras. I.

*(English)*Zbl 0863.17008Let \(L\) 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.

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.

Reviewer: Earl J.Taft (New Brunswick)

##### 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 |

##### Keywords:

quantization of Lie bialgebras; equicontinuous \(G\)-modules; quantum \(R\)-matrix; braided monoidal category; Verma modules; Hopf algebra; quantum double##### References:

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