Twisted Demazure modules, fusion product decomposition and twisted 𝑄-systems

Kus, Deniz; Venkatesh, R.

doi:10.1090/ert/478

Twisted Demazure modules, fusion product decomposition and twisted $Q$-systems
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by Deniz Kus and R. Venkatesh

Represent. Theory 20 (2016), 94-127

DOI: https://doi.org/10.1090/ert/478

Published electronically: February 17, 2016

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Abstract:

In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the twisted current algebras. These modules are indexed by an $|R^+|$-tuple of partitions $\xi =(\xi ^{\alpha })_{\alpha \in R^+}$ satisfying a natural compatibility condition. We give three equivalent presentations of these modules and show that for a particular choice of $\xi$ these modules become isomorphic to Demazure modules in various levels for the twisted affine algebras. As a consequence we see that the defining relations of twisted Demazure modules can be greatly simplified. Furthermore, we investigate the notion of fusion products for twisted modules, first defined by Feigin and Loktev in 1999 for untwisted modules, and use the simplified presentation to prove a fusion product decomposition of twisted Demazure modules. As a consequence we prove that twisted Demazure modules can be obtained by taking the associated graded modules of (untwisted) Demazure modules for simply-laced affine algebras. Furthermore we give a semi-infinite fusion product construction for the irreducible representations of twisted affine algebras. Finally, we prove that the twisted $Q$-sytem defined by Hatayama et al. in 2001 extends to a non-canonical short exact sequence of fusion products of twisted Demazure modules.

References

R. W. Carter, Lie algebras of finite and affine type, Cambridge Studies in Advanced Mathematics, vol. 96, Cambridge University Press, Cambridge, 2005. MR 2188930, DOI 10.1017/CBO9780511614910
Vyjayanthi Chari, Bogdan Ion, and Deniz Kus, Weyl modules for the hyperspecial current algebra, Int. Math. Res. Not. IMRN 15 (2015), 6470–6515. MR 3384485, DOI 10.1093/imrn/rnu135
Vyjayanthi Chari and Adriano Moura, The restricted Kirillov-Reshetikhin modules for the current and twisted current algebras, Comm. Math. Phys. 266 (2006), no. 2, 431–454. MR 2238884, DOI 10.1007/s00220-006-0032-2
Vyjayanthi Chari and Adriano Moura, Kirillov-Reshetikhin modules associated to $G_2$, Lie algebras, vertex operator algebras and their applications, Contemp. Math., vol. 442, Amer. Math. Soc., Providence, RI, 2007, pp. 41–59. MR 2372556, DOI 10.1090/conm/442/08519
Vyjayanthi Chari and Andrew Pressley, Twisted quantum affine algebras, Comm. Math. Phys. 196 (1998), no. 2, 461–476. MR 1645027, DOI 10.1007/s002200050431
Vyjayanthi Chari, Perri Shereen, R. Venkatesh, and Jeffrey Wand, A Steinberg type decomposition theorem for higher level Demazure modules, arXiv:1408.4090.
Vyjayanthi Chari and R. Venkatesh, Demazure modules, fusion products and $Q$-systems, Comm. Math. Phys. 333 (2015), no. 2, 799–830. MR 3296163, DOI 10.1007/s00220-014-2175-x
B. Feigin and E. Feigin, $Q$-characters of the tensor products in $\mathfrak {s}\mathfrak {l}_2$-case, Mosc. Math. J. 2 (2002), no. 3, 567–588. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1988973, DOI 10.17323/1609-4514-2002-2-3-567-588
B. Feigin and S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, vol. 194, Amer. Math. Soc., Providence, RI, 1999, pp. 61–79. MR 1729359, DOI 10.1090/trans2/194/04
Tammy Michelle Fisher-Vasta, Presentations of Z-forms for the universal enveloping algebras of affine Lie algebras, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–University of California, Riverside. MR 2698192
Ghislain Fourier, Tanusree Khandai, Deniz Kus, and Alistair Savage, Local Weyl modules for equivariant map algebras with free abelian group actions, J. Algebra 350 (2012), 386–404. MR 2859894, DOI 10.1016/j.jalgebra.2011.10.018
Ghislain Fourier and Deniz Kus, Demazure modules and Weyl modules: the twisted current case, Trans. Amer. Math. Soc. 365 (2013), no. 11, 6037–6064. MR 3091275, DOI 10.1090/S0002-9947-2013-05846-1
G. Fourier and P. Littelmann, Tensor product structure of affine Demazure modules and limit constructions, Nagoya Math. J. 182 (2006), 171–198. MR 2235341, DOI 10.1017/S0027763000026866
G. Fourier and P. Littelmann, Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), no. 2, 566–593. MR 2323538, DOI 10.1016/j.aim.2006.09.002
Howard Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), no. 2, 480–551. MR 502647, DOI 10.1016/0021-8693(78)90294-6
G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, Remarks on fermionic formula, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998) Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 243–291. MR 1745263, DOI 10.1090/conm/248/03826
Goro Hatayama, Atsuo Kuniba, Masato Okado, Taichiro Takagi, and Zengo Tsuboi, Paths, crystals and fermionic formulae, MathPhys odyssey, 2001, Prog. Math. Phys., vol. 23, Birkhäuser Boston, Boston, MA, 2002, pp. 205–272. MR 1903978
David Hernandez, The Kirillov-Reshetikhin conjecture and solutions of $T$-systems, J. Reine Angew. Math. 596 (2006), 63–87. MR 2254805, DOI 10.1515/CRELLE.2006.052
David Hernandez, Kirillov-Reshetikhin conjecture: the general case, Int. Math. Res. Not. IMRN 1 (2010), 149–193. MR 2576287, DOI 10.1093/imrn/rnp121
A. Joseph, On the Demazure character formula, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 389–419. MR 826100, DOI 10.24033/asens.1493
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
A. N. Kirillov, Identities for the Rogers dilogarithmic function connected with simple Lie algebras, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 164 (1987), no. Differentsial′naya Geom. Gruppy Li i Mekh. IX, 121–133, 198 (Russian, with English summary); English transl., J. Soviet Math. 47 (1989), no. 2, 2450–2459. MR 947332, DOI 10.1007/BF01840426
A. N. Kirillov and N. Yu. Reshetikhin, Representations of Yangians and multiplicities of the inclusion of the irreducible components of the tensor product of representations of simple Lie algebras, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 160 (1987), no. Anal. Teor. Chisel i Teor. Funktsiĭ. 8, 211–221, 301 (Russian); English transl., J. Soviet Math. 52 (1990), no. 3, 3156–3164. MR 906858, DOI 10.1007/BF02342935
Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1923198, DOI 10.1007/978-1-4612-0105-2
Atsuo Kuniba and Junji Suzuki, Functional relations and analytic Bethe ansatz for twisted quantum affine algebras, J. Phys. A 28 (1995), no. 3, 711–722. MR 1326102, DOI 10.1088/0305-4470/28/3/024
Olivier Mathieu, Construction du groupe de Kac-Moody et applications, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 5, 227–230 (French, with English summary). MR 932325
Hiraku Nakajima, $t$-analogs of $q$-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7 (2003), 259–274. MR 1993360, DOI 10.1090/S1088-4165-03-00164-X
Patrick Polo, Variétés de Schubert et excellentes filtrations, Astérisque 173-174 (1989), 10–11, 281–311 (French, with English summary). Orbites unipotentes et représentations, III. MR 1021515
N. Yu. Reshetikhin, The spectrum of the transfer matrices connected with Kac-Moody algebras, Lett. Math. Phys. 14 (1987), no. 3, 235–246. MR 919327, DOI 10.1007/BF00416853
R. Venkatesh, Fusion product structure of Demazure modules, Algebr. Represent. Theory 18 (2015), no. 2, 307–321. MR 3336341, DOI 10.1007/s10468-014-9495-6
Minoru Wakimoto, Lectures on infinite-dimensional Lie algebra, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1873994, DOI 10.1142/9789812810700