Rational smoothness of varieties of representations for quivers of type

Authors:
Robert Bédard and Ralf Schiffler

Journal:
Represent. Theory **7** (2003), 481-548

MSC (2000):
Primary 17B37; Secondary 32S60

DOI:
https://doi.org/10.1090/S1088-4165-03-00179-1

Published electronically:
November 14, 2003

MathSciNet review:
2017066

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, the authors study when the closure (in the Zariski topology) of orbits of representations of quivers of type are rationally smooth. This is done by considering the corresponding quantized enveloping algebra and studying the action of the bar involution on PBW bases. Using Ringel's Hall algebra approach to quantized enveloping algebras and also Auslander-Reiten quivers, we can describe the commutation relations between root vectors. This way we get explicit formulae for the multiplication of an element of PBW bases adapted to a quiver with a root vector and also recursive formulae to study the bar involution on PBW bases. One of the consequences of our characterization is that if the orbit closure is rationally smooth, then it is smooth.

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Additional Information

**Robert Bédard**

Affiliation:
Département de mathematiques, Université du Québec à Montréal, C.P. 8888, Succ. Centre-Ville, Montréal, Québec, H3C 3P8, Canada

Email:
bedard@lacim.uqam.ca

**Ralf Schiffler**

Affiliation:
Département de mathematiques, Université du Québec à Montréal, C.P. 8888, Succ. Centre-Ville, Montréal, Québec, H3C 3P8, Canada

Address at time of publication:
School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottowa, Ontario, K1S 5B6, Canada

Email:
ralf@math.uqam.ca, ralf@math.carleton.ca

DOI:
https://doi.org/10.1090/S1088-4165-03-00179-1

Keywords:
Quantized enveloping algebras,
local intersection cohomology

Received by editor(s):
October 15, 2002

Received by editor(s) in revised form:
July 25, 2003

Published electronically:
November 14, 2003

Additional Notes:
The first author was supported in part by a NSERC grant

The second author was supported in part by a FCAR scholarships

Article copyright:
© Copyright 2003
American Mathematical Society