Journal of the Ramanujan Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

Integrable Irreducible Representations of Toroidal Lie Algebras


Affiliations
1 Institute of Mathematics and Applications, Bhubaneswar, Odisha 751 003, Iceland
     

   Subscribe/Renew Journal


A classification of the irreducible integrable representations with finite-dimensional weight spaces of toroidal Lie algebras was obtained in [R3]. In [CFK], adopting an approach via Weyl modules it was shown that in the case when the central elements act trivially, the results of [R3] hold for any Lie algebra of the form g ⊗ A, where g is a finite-dimensional Lie algebra and A is a finitely generated commutative algebra with unity. In this paper, adopting the approach of [CFK,L] we give a reproof the results of [R3]. In addition we establish a necessary and sufficient condition under which two such irreducible modules are isomorphic.
User
Subscription Login to verify subscription
Notifications
Font Size

  • [AABGP] B. Allison, S. Azam, S. Berman, Y. Gao, A. Pianzola, Extended affine Lie algebras and their ischolar_main systems, Mem. Amer. Math. Soc., 603 (1997) 11–22.

  • [B] P. Batra, Representations of Twisted Multiloop Lie algebras, Journal of Algebra, 272 (2004) 404–416.

  • [PB] P. Batra and T. Pal Representations of Graded Multiloop Lie Algebras, Communications in Algebra, 38(1) 2009 49–67.

  • [BR] P. Batra and S. E. Rao, Classification of irreducible integrable highest weight modules for current KacMoody algebras, Journal of Algebra and its Applications, DOI: http://dx.doi.org/10.1142/S0219498817501237.

  • [Bou1] N. Bourbaki, Elements of Mathematics, Lie Groups and Li Algebras, Chapters 1–3.

  • [Bou4] N. Bourbaki, Elements of Mathematics, Lie Groups and Li Algebras, Chapters 4–6.

  • [C1] V. Chari, Integrable representations of affine Lie-algebras, Invent. Math., 85 (1986) 317-335.

  • [CFK] V. Chari, G. Fourier and T. Khandai, A categorical approach to Weyl modules, Transformation Groups, 15(3) (2010) 517–549.

  • [CG] V. Chari and J. Greenstein, Graded Level Zero Integrable Representations of Affine Lie Algebras, Transactions of AMS., 360 no. 6, (2008) 2923–2940.

  • [CP1] V. Chari and A. Pressley, New unitary representations of loop algebras, Math. Ann., 275 no. 1 (1986), 87–104.

  • [CP2] V. Chari and A. Pressley, A new family of Irreducible, Integrable Modules for Affine Lie algebras, Math. Ann., 277 (1987) 543–562.

  • [CP3] V. Chari and A. Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory, 5 (2001) 191–223 (electronic).

  • [CL] V. Chari and T. Le, Representations of Double Affine Lie Algebras, In a Tribute to C. S. Seshadri (Chennai, 2002), Trends Math., Birkhauser, Basel (2003) 199–219.

  • [G] H. Garland, The arithmetic theory of loop algebras, J. Algebra, 53 (1978) 48–551.

  • [FL] B. Feigin and S. Loktev, Multi-dimensional Weyl modules and symmetric functions, Comm. Math. Phys., 251 (2004) 427–445.

  • [F] V. M. Futorny, Irreducible non-dense A(1)1 modules, Pacific Journal of Mathematics, 172(1) (1996) 83–99.

  • [Kac] V. G. Kac, Infinite dimensional Lie algebras, Prog. Math., Boston 44, Third Edition.

  • [L] M. Lau, Representations of Multiloop Lie algebras, Pacific J. Math., 245(1) (2010) 167–184.

  • [R1] S. E. Rao, Iterated loop modules and a filtration for vertex representation of toroidal Lie algebras, Pacific J. Math., 171(2) (1995) 511–528.

  • [R2] S. E. Rao, Complete reducibility of integrable modules for the affine Lie (super)algebras, Journal of Algebra, 264 no. 1, (2003) 269–278.

  • [R3] S. E. Rao, Classification of irreducible integrable modules for toroidal Lie algebras with finite dimensional weight spaces, Journal of Algebra, 277 (2004) 318–348.

  • [R4] S. E. Rao, Irreducible Representations for toroidal Lie algebras, Journal of Pure and Applied Algebra, 202(1–3) (2005) 102–117.

  • [RM] S. E. Rao and R.V. Moody, Vertex Representations for N-Toroidal Lie algebras and a Generalization of the Virasoro Algebras, Communications in Mathematical Physics, 159 (1994) 239–264.

  • [RFS] S. E. Rao, S. Sharma and V. Futorny, Weyl Modules associated to Kac-Moody Lie Algebras, Communications in Algebra, 44:12, 5045–5057, DOI:10.1080/00927872.2015.1130143.

  • [VV] R. Venkatesh and S. Viswanath, Unique factorization of tensor products for Kac-Moody algebras, Adv. Math., 231 no. 6, (2012) 3162–3171.


Abstract Views: 249

PDF Views: 0




  • Integrable Irreducible Representations of Toroidal Lie Algebras

Abstract Views: 249  |  PDF Views: 0

Authors

Tanusree Khandai
Institute of Mathematics and Applications, Bhubaneswar, Odisha 751 003, Iceland

Abstract


A classification of the irreducible integrable representations with finite-dimensional weight spaces of toroidal Lie algebras was obtained in [R3]. In [CFK], adopting an approach via Weyl modules it was shown that in the case when the central elements act trivially, the results of [R3] hold for any Lie algebra of the form g ⊗ A, where g is a finite-dimensional Lie algebra and A is a finitely generated commutative algebra with unity. In this paper, adopting the approach of [CFK,L] we give a reproof the results of [R3]. In addition we establish a necessary and sufficient condition under which two such irreducible modules are isomorphic.

References