Modules of the toroidal Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$

Document Type: Research Paper

Authors

1 Department of Mathematics, ‎Shanghai University, ‎Shanghai 200444‎, ‎China.

2 School of Information and Mathematics, ‎Yangtze University, ‎Jingzhou 434023‎, ‎China.

Abstract

‎Highest weight modules of the double affine Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎Singular vectors of Verma modules are‎ ‎determined using a similar condition with horizontal affine Lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

Keywords

Main Subjects


S. Berman and Y. Billig, Irreducible representations for toroidal Lie algebras, J. Algebra 221 (1999), no. 1, 188--231.

V. Chari, Representations of affine and toroidal Lie algebras, Geometric representation theory and extended affine Lie algebras, pp. 169--197, Fields Inst. Commun. 59, Amer. Math. Soc. Providence, RI, 2011.

V. Chari and T. Le, Representations of double affine Lie algebras, A tribute to C. S. Seshadri (Chennai, 2002), pp. 199--219, Trends Math. Birkhäuser, Basel, 2003.

X. Chang and S. Tan, A class of irreducible integarable modules for the extended baby TKK algebra, Pacific J. Math. 252 (2011), no. 2, 293--312.

S. Eswara Rao, Classification of irreducible integrable modules for multi-loop algebras with finite dimensional weight spaces, J. Algebra 246 (2001) 215--225.

S. Eswara Rao, Classification of irreducible integrable modules for toroidal Lie algebras with finite dimensional weight spaces, J. Algebra 277 (2004), no. 1, 318--348.

S. Eswara Rao, On representations of toroidal Lie algebras, Functional analysis VIII, pp. 146--167, Various Publ. Ser. (Aarhus), 47, Aarhus Univ. Aarhus, 2004.

[8] S. Eswara Rao, Irreducible representations for toroidal Lie algebras, J. Pure Appl. Algebra 202 (2005), no. 1-3, 102--117.

S. Eswara Rao and C. Jiang, Classification of irreducible integrable representations for the full toroidal Lie algebras, J. Pure Appl. Algebra 200 (2005), no. 1, 71--85.

V. E. Futorny, Imaginary Verma modules for affine Lie algebras, Canad. Math. Bull. 37 (1994), no. 2, 213--218.

N. Jing and K. C. Misra, Fermionic realization of toroidal Lie algebras of classical types, J. Algebra 324 (2010), no. 2, 183--194.

V. G. Kac, Infinite Dimensional Lie Algebras, Cambridge University Press, 3rd edition, 1990.

V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite dimensional Lie algebra, Adv. Math. 34 (1979), no. 1, 97--108.

F. G. Malikov, B. L. Feigin and D. B. Fuks, Singular vectors in Verma modules over Kac-Moody algebras, Funct. Anal. Appl. 20 (1986), no. 2, 103--113.

E. V. Moody and Z. Shi, Toroidal Weyl groups, Nova J. Algebra Geom. 1 (1992), no. 4, 317--337.