Sur une algèbre Q-symétrique

A. Guichardet

Annales Polonici Mathematici (1997)

  • Volume: 66, Issue: 1, page 123-135
  • ISSN: 0066-2216

Abstract

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We establish several properties of a quadratic algebra over a field k, which is a deformation of the symmetric algebra Sk³. In particular, we prove that A is an integral domain, noetherian and Koszul; we compute its first Hochschild cohomology group; we determine the corresponding Poisson structure on k³ and its symplectic leaves; we define an involution on A and describe the corresponding irreducible involutive representations.

How to cite

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A. Guichardet. "Sur une algèbre Q-symétrique." Annales Polonici Mathematici 66.1 (1997): 123-135. <http://eudml.org/doc/270022>.

@article{A1997,
abstract = {We establish several properties of a quadratic algebra over a field k, which is a deformation of the symmetric algebra Sk³. In particular, we prove that A is an integral domain, noetherian and Koszul; we compute its first Hochschild cohomology group; we determine the corresponding Poisson structure on k³ and its symplectic leaves; we define an involution on A and describe the corresponding irreducible involutive representations.},
author = {A. Guichardet},
journal = {Annales Polonici Mathematici},
keywords = {deformations; derivations; symplectic leaves; representations; quadratic algebras; deformations of symmetric algebras; Hochschild cohomology groups; Poisson structures; irreducible involutive representations},
language = {fre},
number = {1},
pages = {123-135},
title = {Sur une algèbre Q-symétrique},
url = {http://eudml.org/doc/270022},
volume = {66},
year = {1997},
}

TY - JOUR
AU - A. Guichardet
TI - Sur une algèbre Q-symétrique
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 123
EP - 135
AB - We establish several properties of a quadratic algebra over a field k, which is a deformation of the symmetric algebra Sk³. In particular, we prove that A is an integral domain, noetherian and Koszul; we compute its first Hochschild cohomology group; we determine the corresponding Poisson structure on k³ and its symplectic leaves; we define an involution on A and describe the corresponding irreducible involutive representations.
LA - fre
KW - deformations; derivations; symplectic leaves; representations; quadratic algebras; deformations of symmetric algebras; Hochschild cohomology groups; Poisson structures; irreducible involutive representations
UR - http://eudml.org/doc/270022
ER -

References

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  2. [2] A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473-527. Zbl0864.17006
  3. [3] E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, The 3-dimensional Euclidean quantum group E ( 3 ) q and its R-matrix, J. Math. Phys. 32 (1991), 1159-1165. Zbl0743.17017
  4. [4] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, 1994. Zbl0839.17009
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  6. [6] C. Kassel, Quantum Groups, Springer, 1995. 
  7. [7] R. K. Molnar, Semi-direct products of Hopf algebras, J. Algebra 47 (1977), 29-51. Zbl0353.16004
  8. [8] Ya. S. Soĭbel'man, The algebra of functions on a compact quantum group and its representations, Algebra i Analiz 2 (1) (1990), 190-212 (in Russian); English transl.: Leningrad Math. J. 2 (1) (1991), 161-178. Zbl0708.46029
  9. [9] L. L. Vaksman and Y. S. Soĭbel'man, An algebra of functions on the quantum group SU(2), Funktsional. Anal. i Prilozhen. 22 (3) (1988), 1-14 (in Russian); English transl.: Funct. Anal. Appl. 22 (3) (1988), 170-181. Zbl0661.43001
  10. [10] M. Van den Bergh, Noncommutative homology of some 3-dimensional quantum spaces, K-Theory 8 (1994), 213-230. 
  11. [11] M. Wambst, Complexes de Koszul quantiques, Ann. Inst. Fourier (Grenoble) 43 (1993), 1089-1156. Zbl0810.16010
  12. [12] S. L. Woronowicz, Quantum E(2) group and its Pontryagin dual, Lett. Math. Phys. 23 (1991), 251-263. Zbl0752.17017

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