Semicanonical bases arising from enveloping algebras. (English) Zbl 0983.17009

Lusztig, G.

Adv. Math. 151, No. 2, 129-139 (2000).

Let \(U^+\) be the plus part of the enveloping algebra of a Kac-Moody Lie algebra \({\mathfrak g}\) with a symmetric Cartan datum. In [G. Lusztig, J. Am. Math. Soc. 3, 447-498 (1990; Zbl 0703.17008)] a canonical basis of \(U^+\) is given under the assumption that the Cartan datum is of finite type. The basis of \(U^+\) is obtained from a canonical basis of the quantized version of \(U^+\) by specializing the quantum parameter to 1. In [G. Lusztig, Publ. Math., Inst. Hautes Étud. Sci. 76, 111-163 (1992; Zbl 0776.17013)] a basis of \(U^+\) is constructed in terms of constructible functions on a Lagrangian variety with the Cartan datum of affine type, which is called the semicanonical basis of \(U^+\).
In this paper, the author extends the definition of semicanonical basis to include the case where the Cartan datum is not necesarily of affine or finite type. When the semicanonical basis is not necessarily the same as the canonical basis, it was shown that the semicanonical basis has a number of properties in common with the canonical basis: compatibility with various filtrations of \(U^+\), compatibility with the canonical antiautomorphism of \(U^+\).

Reviewer: Fang Li (Hangzhou)

MSC:

17B35	Universal enveloping (super)algebras
17B67	Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
16S30	Universal enveloping algebras of Lie algebras

Keywords:

enveloping algebra; Kac-Moody Lie algebra; semicanonical basis; Cartan datum; canonical basis

References:

[1]	Kashiwara, M., On crystal bases of the q-analogue of universal enveloping algebras, Duke math. J., 63, 465-516, (1991) · Zbl 0739.17005
[2]	Kashiwara, M.; Saito, Y., Geometric construction of crystal bases, Duke math. J., 89, 9-36, (1997) · Zbl 0901.17006
[3]	Lusztig, G., Canonical bases arising from quantized enveloping algebras, J. amer. math. soc., 3, 447-498, (1990) · Zbl 0703.17008
[4]	Lusztig, G., Canonical bases arising from quantized enveloping algebras, II, Progr. theoret. phys. suppl., 102, 175-201, (1990) · Zbl 0776.17012
[5]	Lusztig, G., Quivers, perverse sheaves and enveloping algebras, J. amer. math. soc., 4, 365-421, (1991) · Zbl 0738.17011
[6]	Lusztig, G., Inst. hautes études sci. publ. math., 76, 111-163, (1992)
[7]	Nakajima, H., Instantons on ALE spaces, quiver varieties and Kac-Moody algebras, Duke math. J., 76, 365-416, (1994) · Zbl 0826.17026