Structure theorems of E(n)-Azumaya algebras

Abstract

Let k be a field and E(n) be the 2ⁿ⁺¹-dimensional pointed Hopf algebra over k constructed by Beattie, Dăscălescu and Grünenfelder [J. Algebra, 2000, 225: 743–770]. E(n) is a triangular Hopf algebra with a family of triangular structures R_M parameterized by symmetric matrices M in M_n(k). In this paper, we study the Azumaya algebras in the braided monoidal category $E_{(n)} \mathcal{M}^{R_M } $ and obtain the structure theorems for Azumaya algebras in the category $E_{(n)} \mathcal{M}^{R_M } $, where M is any symmetric n×n matrix over k.

Keywords

Yetter-Drinfeld module / Brauer group / Azumaya algebra

References

[1] Armour A., Chen H. X., Zhang Y. H. Structure theorems of H₄-Azumaya algebras. J Algebra, 2006, 305: 360-393.

[2] Beattie M., Dăscălescu S., Grünenfelder L. Constructing pointed Hopf algebras by Ore extension. J Algebra, 2000, 225: 743-770.

[3] Caenepeel S., Van Oystaeyen F., Zhang Y. H. Quantum Yang-Baxter module algebras. K-Theory, 1993, 8: 231-255.

[4] Caenepeel S., Van Oystaeyen F., Zhang Y. H. The Brauer group of Yetter-Drinfeld module algebras. Trans Amer Math Soc, 1997, 349: 3737-3771.

[5] Carnovale G. Some isomorphisms for the Brauer groups of a Hopf algebra. Comm Algebra, 2001, 29: 5291-5305.

[6] Carnovale G., Cuadra J. Cocycle twisting of E(_n)-module algebras and applications to the Brauer group. K-Theory, 2004, 33: 251-276.

[7] Chen H. X., Zhang Y. H. Cocycle deformations and isomorphisms of Brauer groups. Comm Algebra, 2007, 35: 399-433.

[8] DeMeyer F., Ford T. Computing the Brauer group of ℤ2-dimodule algebras. J Pure Appl Algebra, 1988, 54: 197-208.

[9] Jacobson N. Basic Algebra II, 1980, San Francisco: Freeman WH and Company.

[10] Kassel C. Quantum Groups, 1995, New York: Springer-Verlag.

[11] Lam T. Y. The Algebraic Theory of Quadratic Forms, 1973, San Francisco: The Benjamin/Cummings Publishing Company, INC.

[12] Majid S. Foundations of Quantum Group Theory, 1995, Cambridge: Cambridge Univ Press

[13] Montgomery S. Hopf Algebras and Their Actions on Rings. CBMS Series in Math, 1993, Providence: Amer Math Soc.

[14] Panaite F., Van Oystaeyen F. Quasitriangular structures for some pointed Hopf algebras of dimension 2ⁿ. Comm Algebra, 1999, 27: 4929-4942.

[15] Small C. The Brauer-Wall group of a commutative ring. Trans Amer Soc, 1971, 156: 455-491.

[16] Sweedler M. E. Hopf Algebras, 1969, New York: Benjamin.

[17] Van Oystaeyen F., Zhang Y. H. The Brauer group of a braided monoidal category. J Algebra, 1998, 202: 96-128.

[18] Van Oystaeyen F., Zhang Y. H. The Brauer group of a Hopf algebra. New Directions in Hopf Algebras. MSRI Publications, 2002, 43: 437-485.

[19] Wall C. T. C. Graded Brauer groups. J Reine Angew Math, 1964, 213: 187-199.