The bar automorphism in quantum groups and geometry of quiver representations

Philippe Caldero[1]; Markus Reineke[2]

  • [1] Université Claude Bernard Lyon I Département de mathématiques 69622 Villeurbanne Cedex (France)
  • [2] Universität Münster Mathematisches Institut 48149 Münster (Germany)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 1, page 255-267
  • ISSN: 0373-0956

Abstract

top
Two geometric interpretations of the bar automorphism in the positive part of a quantized enveloping algebra are given. The first is in terms of numbers of rational points over finite fields of quiver analogues of orbital varieties; the second is in terms of a duality of constructible functions provided by preprojective varieties of quivers.

How to cite

top

Caldero, Philippe, and Reineke, Markus. "The bar automorphism in quantum groups and geometry of quiver representations." Annales de l’institut Fourier 56.1 (2006): 255-267. <http://eudml.org/doc/10141>.

@article{Caldero2006,
abstract = {Two geometric interpretations of the bar automorphism in the positive part of a quantized enveloping algebra are given. The first is in terms of numbers of rational points over finite fields of quiver analogues of orbital varieties; the second is in terms of a duality of constructible functions provided by preprojective varieties of quivers.},
affiliation = {Université Claude Bernard Lyon I Département de mathématiques 69622 Villeurbanne Cedex (France); Universität Münster Mathematisches Institut 48149 Münster (Germany)},
author = {Caldero, Philippe, Reineke, Markus},
journal = {Annales de l’institut Fourier},
keywords = {quantum groups; quiver representations; bar automorphism; preprojective variety; canonical basis},
language = {eng},
number = {1},
pages = {255-267},
publisher = {Association des Annales de l’institut Fourier},
title = {The bar automorphism in quantum groups and geometry of quiver representations},
url = {http://eudml.org/doc/10141},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Caldero, Philippe
AU - Reineke, Markus
TI - The bar automorphism in quantum groups and geometry of quiver representations
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 1
SP - 255
EP - 267
AB - Two geometric interpretations of the bar automorphism in the positive part of a quantized enveloping algebra are given. The first is in terms of numbers of rational points over finite fields of quiver analogues of orbital varieties; the second is in terms of a duality of constructible functions provided by preprojective varieties of quivers.
LA - eng
KW - quantum groups; quiver representations; bar automorphism; preprojective variety; canonical basis
UR - http://eudml.org/doc/10141
ER -

References

top
  1. K. Bongartz, On degenerations and extensions of finite dimensional modules, Adv. Math. 121 (1996), 245-287 Zbl0862.16007MR1402728
  2. P. Caldero, A multiplicative property of quantum flag minors, Representation Theory 7 (2003), 164-176 Zbl1030.17009MR1973370
  3. P. Caldero, R. Schiffler, Rational smoothness of varieties of representations for quivers of Dynkin type, Ann. Inst. Fourier (Grenoble) 54 (2004), 295-315 Zbl1126.17013MR2073836
  4. D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184 Zbl0499.20035MR560412
  5. G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498 Zbl0703.17008MR1035415
  6. G. Lusztig, Canonical bases arising from quantized enveloping algebras II, Common trends in mathematics and quantum field theories (1990), 175-201, EguchiT.T. Zbl0776.17012MR1182165
  7. G. Lusztig, Introduction to quantum groups, 110 (1993), Birkhäuser, Boston Zbl0788.17010MR1227098
  8. M. Reineke, Multiplicative properties of dual canonical bases of quantum groups, J. Algebra 211 (1999), 134-149 Zbl0917.17008MR1656575
  9. C. Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994), 526-546 Zbl0841.16018MR1302854
  10. C. M. Ringel, Tame algebras and integral quadratic forms, 1099 (1984), Springer, Berlin Zbl0546.16013MR774589
  11. C. M. Ringel, Hall algebras, Topics in Algebra, Part I 26 (1990), 433-447, Warsaw, 1988 Zbl0778.16004MR1171248
  12. C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), 583-591 Zbl0735.16009MR1062796

NotesEmbed ?

top

You must be logged in to post comments.