## Rational smoothness of varieties of representations for quivers of type $A$

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- by Robert Bédard and Ralf Schiffler
- Represent. Theory
**7**(2003), 481-548 - DOI: https://doi.org/10.1090/S1088-4165-03-00179-1
- Published electronically: November 14, 2003
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## Abstract:

In this paper, the authors study when the closure (in the Zariski topology) of orbits of representations of quivers of type $A$ are rationally smooth. This is done by considering the corresponding quantized enveloping algebra ${\mathbf {U}}$ 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.## References

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*Rational smoothness of varieties of representations for quivers of Dynkin type*, preprint.

## Bibliographic 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
- MR Author ID: 724459
- Email: ralf@math.uqam.ca, ralf@math.carleton.ca
- 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 - © Copyright 2003 American Mathematical Society
- Journal: Represent. Theory
**7**(2003), 481-548 - MSC (2000): Primary 17B37; Secondary 32S60
- DOI: https://doi.org/10.1090/S1088-4165-03-00179-1
- MathSciNet review: 2017066