Dr Robert Marsh (University of Leicester)

Regions of the canonical basis of a quantized enveloping algebra.

This talk is on joint work with Markus Reineke (Wuppertal). The quantized enveloping algebra associated to a finite-dimensional simple Lie algebra over the complex numbers (such as the n by n matrices of trace zero) is a deformed version of its universal enveloping algebra which allows a ”canonical basis” with many useful properties. This basis can be specialised back to the universal enveloping algebra to give a basis which has no known direct construction.

The canonical basis has many different interpretations — for example, in terms of the representation theory of affine Hecke algebras [2], which are related to its division into regions. These regions also have fascinating combinatorial properties [1]. In this talk we will look at how the connection between the canonical basis and the preprojective algebra [3], [4], can be used to give a description of regions of the canonical basis in terms of the geometric representation theory of the preprojective algebra.

References

[1] R. Carter and R. Marsh. Regions of linearity, Lusztig cones and canonical basis elements for the quantized enveloping algebra of type A_{4}. J. Algebra 234, No.2, 2000, 545–603.

[2] B. Leclerc, M. Nazarov and J.-Y. Thibon. Induced representations of affine Hecke algebras and canonical bases of quantum groups. Preprint arXiv:math.QA/0011074, 2000.

[3] M. Kashiwara and Y. Saito. Geometric construction of crystal bases. Duke Math. J. 89, No.1, 1997, 9–36.

[4] G. Lusztig. Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc. 3, No.2, 1990, 447–498.