Generalized Verma modules over some Block algebras

Abstract

In this paper, a class of generalized Verma modules M(V) over some Block type Lie algebra ℬ(G) are constructed, which are induced from nontrivial simple modules V over a subalgebra of ℬ(G). The irreducibility of M(V) is determined.

Keywords

Lie algebra of Block type / generalized Verma module / irreducible module

References

[1] Block R. On torsion-free abelian groups and Lie algebras. Proc Amer Math Soc, 1958, 9: 613-620.

[2] Hu J., Wang X. D., Zhao K. M. Verma modules over generalized Virasoro algebras Vir[G]. J Pure Appl Algebra, 2003, 177: 61-69.

[3] Kac V., Radul A. Quasi-finite highest weight modules over the Lie algebra of differential operators on the circle. Comm Math phys, 1993, 157: 429-457.

[4] Kac V., Raina A. Highest Weight Representations of Infinite Dimensional Lie Algebras, 1987, Hong Kong: World Scientific, 1-9.

[5] Khomenko A., Mazorchuk V. Generalized Verma modules over the Lie algebra of type G₂. Comm Algebra, 1999, 27: 777-783.

[6] Khomenko A., Mazorchuk V. Generalized Verma modules induced from sl(2,ℂ) and associated Verma modules. J Algebra, 2001, 242(2): 561-576.

[7] Mazorchuk V. Verma modules over generalized Witt algebra. Compos Math, 1999, 115(1): 21-35.

[8] Mazorchuk V. The Structure of Generalized Verma Modules. Dissertation for the Doctoral Degree. Kyiv University, 1996, 5–12

[9] Su Y. Classification of quasifinite modules over the Lie algebras of Weyl type. Adv Math, 2003, 174: 57-68.

[10] Su Y. Quasifinite representations of a Lie algebra of Block type. J Algebra, 2004, 276: 117-128.

[11] Su Y. Quasifinite representations of a family of Lie algebras of Block type. J Pure Appl Algebra, 2004, 192: 293-305.

[12] Verma D. Structure of certain induced representations of complex semisimple Lie algebras. Bull Amer Math Soc, 1968, 74: 160-166.

[13] Wang X., Zhao K. Verma modules over the Virasoro-like algebra. J Australia Math, 2006, 80: 179-191.

[14] Wu Y., Su Y. Highest weight representations of a Lie algebra of Block type. Science in China, Ser A, 2007, 50(4): 549-560.

[15] Xu X. Generalizations of Block algebras. Manuscripta Math, 1999, 100: 489-518.

[16] Xu X. Quadratic conformal superalgebras. J Algebra, 2000, 231: 1-38.

[17] Xu X. New generalized simple Lie algebras of Cartan type over a field with characteristic 0. J Algebra, 2000, 224: 23-58.

[18] Yue X, Su Y, Xin B. Highest weight representations of a family of Lie algebras of Block type. Acta Mathematica Sinica (English Ser) (in press)

[19] Zhang C, Su Y. Generalized Verma module over nongraded Witt algebra. J Univ Sci Tech China (in press)