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**Quivers, perverse sheaves, and quantized enveloping algebras.**
*(English)*
Zbl 0738.17011

In earlier work [J. Am. Math. Soc. 3, 447-498 (1990; Zbl 0703.17008) and Prog. Theor. Phys. 102, Suppl., 175-202 (1990; Zbl 0776.17012)] the author constructed a very remarkable so-called canonical basis for \(U^ -_ q\) where \(U_ q\) is the quantized enveloping algebra for a semisimple Lie algebra of simply laced type. In fact, he had two different methods for this construction, an elementary and a geometric method.

In this paper he extends the geometric method (using representations of quivers, perverse sheaves etc.) to the case of quantum groups associated with arbitrary Kac-Moody algebras. Among the amazing properties of the canonical basis of \(U_ q\) are its integrality properties, positivity properties and the fact that it gives canonical bases for all the integrable simple modules. All these properties are proved to hold also in the general case (in fact even stronger positivity results than in earlier work are obtained).

One of the key ingredients in the proofs is an imitation of the methods from the author’s theory of character sheaves.

In this paper he extends the geometric method (using representations of quivers, perverse sheaves etc.) to the case of quantum groups associated with arbitrary Kac-Moody algebras. Among the amazing properties of the canonical basis of \(U_ q\) are its integrality properties, positivity properties and the fact that it gives canonical bases for all the integrable simple modules. All these properties are proved to hold also in the general case (in fact even stronger positivity results than in earlier work are obtained).

One of the key ingredients in the proofs is an imitation of the methods from the author’s theory of character sheaves.

Reviewer: H.H.Andersen (Aarhus)

### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

20G05 | Representation theory for linear algebraic groups |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

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\textit{G. Lusztig}, J. Am. Math. Soc. 4, No. 2, 365--421 (1991; Zbl 0738.17011)

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### References:

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