Canonical bases arising from quantized enveloping algebras. II.

*(English)*Zbl 0776.17012This paper is a continuation of the author’s paper [J. Am. Math. Soc. 3, 447-498 (1990; Zbl 0703.17008)] in which the so-called canonical bases for finite-dimensional simple modules for (simply laced) quantum groups were defined. These bases have rather remarkable properties and in the present paper some further extremely interesting results about these bases are obtained.

The author first compares his definition of canonical bases with a recent definition due to Kashiwara of so-called crystal bases (the word crystal basis was already used by M. Kashiwara in [Commun. Math. Phys. 133, 249-260 (1990; Zbl 0724.17009)] but only in [Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)] he did prove their “global” nature).

The most amazing (at least to this reviewer) property of this basis for the quantized enveloping algebra of a semi-simple Lie algebra is the fact that it simultaneously gives rise to bases for all the finite-dimensional simple modules. By specializing the quantum parameter to 1 one gets even bases for the ordinary simple modules for the complex Lie algebra in question. In the present paper it is proved that one can go further and obtain bases for the coinvariants in triple tensor products. One of the nice consequences of this results is (as observed by S. Donkin) that it can be used to give a short proof of the fact that tensor products of Weyl modules for semi-simple algebraic groups have Weyl filtrations. For details as well as a quantized version of this application of canonical bases see the paper by Paradowski in [Proc. Symp. Pure Math. (to appear)].

Another application given in the paper under review is a new combinatorial character formula for the irreducible characters of quantum groups (and hence also of semi-simple Lie algebras).

Throughout the paper it is assumed that the root system in question is simply laced. However, it is indicated that this assumption is not necessary and in his recent book [Introduction to Quantum Groups (Prog. Math. Vol. 110) (BirkhĂ¤user 1993)] the author has shown how to handle also general root systems (including to some extent affine systems as well).

The author first compares his definition of canonical bases with a recent definition due to Kashiwara of so-called crystal bases (the word crystal basis was already used by M. Kashiwara in [Commun. Math. Phys. 133, 249-260 (1990; Zbl 0724.17009)] but only in [Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)] he did prove their “global” nature).

The most amazing (at least to this reviewer) property of this basis for the quantized enveloping algebra of a semi-simple Lie algebra is the fact that it simultaneously gives rise to bases for all the finite-dimensional simple modules. By specializing the quantum parameter to 1 one gets even bases for the ordinary simple modules for the complex Lie algebra in question. In the present paper it is proved that one can go further and obtain bases for the coinvariants in triple tensor products. One of the nice consequences of this results is (as observed by S. Donkin) that it can be used to give a short proof of the fact that tensor products of Weyl modules for semi-simple algebraic groups have Weyl filtrations. For details as well as a quantized version of this application of canonical bases see the paper by Paradowski in [Proc. Symp. Pure Math. (to appear)].

Another application given in the paper under review is a new combinatorial character formula for the irreducible characters of quantum groups (and hence also of semi-simple Lie algebras).

Throughout the paper it is assumed that the root system in question is simply laced. However, it is indicated that this assumption is not necessary and in his recent book [Introduction to Quantum Groups (Prog. Math. Vol. 110) (BirkhĂ¤user 1993)] the author has shown how to handle also general root systems (including to some extent affine systems as well).

Reviewer: H.H.Andersen (Aarhus)

##### MSC:

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

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

20G05 | Representation theory for linear algebraic groups |