Singular vectors of quantum group representations for straight Lie algebra roots

✍ Scribed by V. K. Dobrev

• Publisher: Springer
• Year: 1991
• Tongue: English
• Weight: 689 KB
• Volume: 22
• Category: Article
• ISSN: 0377-9017
• DOI: 10.1007/bf00405600

✦ Synopsis

We give explicit formulae for singular vectors of Verma modules over Uq (fr where f# Is any complex simple Lie algebra. The vectors we present correspond exhaustively to a class of positive roots of <5 which we call straight roots. In some special cases, we give singular vectors corresponding to arbitrary positive roots. For our vectors we use a special basis of U~(C~-), where c~ is the negative roots subalgebra of fr which was introducted in our earlier work in the case q = 1. This basis seems more economical than the PoincarS-Birkhoff-Witt type of basis used by Malikov, Feigin, and Fuchs for the construcuon of singular vectors of Verma modules in the case q = 1. Furthermore, this basis turns out to be part of a general basis recently introduced for other reasons by Lusztig for Uq(~ -), where ,~ is a Borel subalgebra of ~.