No Access

A MATRIX REALIZATION OF THE QUANTUM GROUP 𝔤_p,q

YUSUKE ARIKE

Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan

Search for more papers by this author

https://doi.org/10.1142/S0129167X11006805Cited by:0 (Source: Crossref)

Abstract

In this paper we shall find a matrix realization of the quantum group 𝔤_p,q introduced in [B. L. Feigin et al., Nucl. Phys. B757 (2006) 303–343]. For this purpose, we construct all primitive idempotents and a basis of 𝔤_p,q. We determine the action of elements of the basis on indecomposable projective modules, that gives rise to a matrix realization of 𝔤_p,q. By using this result, we obtain a basis of the space of symmetric linear functions on 𝔤_p1,p2 and express the symmetric linear functions obtained by the left integral, the balancing element, and the center of 𝔤_p,q in terms of this basis.

Keywords:

AMSC: 16W35, 17B37, 81R05

References

Y. Arike, Osaka. J. Math. 47, 535 (2010). Web of Science, Google Scholar
B. L. Feiginet al., Commun. Math. Phys. 265, 47 (2006), DOI: 10.1007/s00220-006-1551-6. Crossref, Web of Science, Google Scholar
B. L. Feiginet al., Theor. Math. Phys. 148, 1210 (2006). Web of Science, Google Scholar
B. L. Feiginet al., Nucl. Phys. B 757, 303 (2006), DOI: 10.1016/j.nuclphysb.2006.09.019. Crossref, Web of Science, Google Scholar
B. L. Feiginet al., J. Math. Phys. 48, 032303 (2007), DOI: 10.1063/1.2423226. Crossref, Web of Science, Google Scholar
C. Kassel , Quantum Groups , Graduate Text in Mathematics 155 ( Springer-Verlag , New York , 1995 ) . Crossref, Google Scholar
H. Kondo and Y. Saito, Indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to $\boldsymbol{\mathfrak{sl}_2}$, preprint , arXiv:0901.4221v2 . Google Scholar
G. Lusztig, Adv. Math. 70(2), 237 (1988), DOI: 10.1016/0001-8708(88)90056-4. Crossref, Web of Science, Google Scholar
S. Montgomery , Hopf Algebras and Their Actions on Rings , CBMS Conf. Series in Math. 82 ( Amer. Math. Soc. , Providence, RI , 1993 ) . Crossref, Google Scholar
K. Nagatomo and A. Tsuchiya, The triplet vertex operator algebra W(p) and the restricted quantum group at root of unity, preprint , arXiv:0902.4607 . Google Scholar
D. E. Radford, J. Algebra 163, 583 (1994), DOI: 10.1006/jabr.1994.1033. Crossref, Web of Science, Google Scholar
R. Suter, Commun. Math. Phys. 163, 359 (1994), DOI: 10.1007/BF02102012. Crossref, Web of Science, Google Scholar