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

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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

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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

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  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

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