Classification of quasifinite modules with nonzero central charges for EALAs of type A with coordinates in quantum torus

Abstract

The author first constructs a Lie algebra $\mathfrak{L}: = \mathfrak{L}(q,w_d )$ from rank 3 quantum torus, which is isomorphic to the core of EALAs of type A _d−1 with coordinates in quantum torus C _q ^d, and then gives the necessary and sufficient conditions for the highest weight modules to be quasifinite. Finally the irreducible ℤ-graded quasifinite $\mathfrak{L}$-modules with nonzero central charges are classified.

Keywords

Core of EALAs / Graded modules / Quasifinite module / Highest weight module / Quantum torus

References

[1] Hoegh-Krohn R., Torresani B.. Classification and construction of quasi-simple Lie algebra. J. Funct. Anal., 1990, 89: 106-136

[2] Berman S., Gao Y., Krylyuk Y.. Quantum tori and the structure of elliptic quasi-simple Lie algebras. J. Funct. Anal., 1996, 135: 339-389

[3] Gao Y., Zeng Z.. Hermitian representations of the extended affine Lie algebra $\widetilde{gl_2 (C_\mathcal{Q} )}$. Adv. Math., 2006, 207: 244-265

[4] Moody R. V., Rao S. E., Yokonuma T.. Toroidal Lie algebras and vertex representations. Geom. Dedecata, 1990, 35: 283-307

[5] Yamada H.. Extended affine Lie algebras and their vertex representations. Publ. Res. Inst. Math. Sci., 1989, 25: 587-603

[6] Berman S., Cox B.. Enveloping algebras and representations of toroidal Lie algebras. Pacific J. Math., 1996, 165: 339-389

[7] Berman S., Gao Y., Krylyuk Y. The alternative torus and the structure of elliptic quasi-simple Lie algebras of type A ₂. Trans. Amer. Math. Soc., 1995, 347: 4315-4363

[8] Manin Y. I.. Topics in Noncommutative Geometry, 1991, Princeton: Princeton University Press

[9] Kirkman E., Procesi C., Small L.. A q-analog for the Virasoro algebra. Comm. Algebra, 1994, 22: 3755-3774

[10] Berman S., Szmigielski J.. Principal realization for extended affine Lie algebra of type sl₂ with coordinates in a simple quantum torus with two variables. Contemp. Math., 1999, 248: 39-67

[11] Gao Y.. Vertex operators arising from the homogeneous realization for $\widehat{gl}_N $. Comm. Math. Phys., 2000, 211: 745-777

[12] Gao Y.. Representations of extended affine Lie algebras coordinatized by certain quantum tori. Comp. Math., 2000, 123: 1-25

[13] Eswara Rao S.. A class of integrable modules for the core of EALA coordinatized by quantum tori. J. Algebra, 2004, 275: 59-74

[14] Eswara Rao S.. Unitary modules for EALAs coordinatized by a quantum torus. Comm. Algebra, 2003, 31: 2245-2256

[15] Zhang H. C., Zhao K. M.. Representations of the Virasoro-like Lie algebras and its q-analog. Comm. Algebra, 1996, 24: 4361-4372

[16] Jiang C. P., Meng D. J.. The derivation algebra of the associative algebra C _q[X, Y,X ₁, Y ₁]. Comm. Algebra, 1998, 26: 1723-1736

[17] Eswara Rao S., Zhao K. M.. Highest weight irreducible representations of rank 2 quantum tori. Math. Res. Lett., 2004, 11: 615-628

[18] Lin W. Q., Tan S. B.. Nonzero level Harish-Chandra modules over the q-analog of the Virasoro-like algebra. J. Algebra, 2006, 297: 254-272

[19] Herstein I. N.. On the Lie and Jordan rings of a simple associative ring. Amer. J. Math., 1955, 77: 279-285

[20] Zhao K. M.. The q-Virasoro-like algebra. J. Algebra, 1997, 188(2): 506-512

[21] Billig Y., Zhao K. M.. Weight modules over exp-polynomial Lie algebras. J. Pure Appl. Algebra, 2004, 191: 23-42

[22] Lü, R. C. and Zhao, K. M., Verma modules over quantum torus Lie algebras, Canadian J. Math., to appear.

[23] Lin, W. Q. and Su, Y. C., Classification of quasifinite representations with nonzero central charges for type A ₁ EALA with coordinates in quantum torus, preprint, 2007. arXiv:0705.4539v1