Quasi-finite modules for Lie superalgebras of infinite rank

Lam, Ngau; Zhang, R.

doi:10.1090/S0002-9947-05-03795-5

Quasi-finite modules for Lie superalgebras of infinite rank
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by Ngau Lam and R. B. Zhang PDF

Trans. Amer. Math. Soc. 358 (2006), 403-439 Request permission

Abstract:

We classify the quasi-finite irreducible highest weight modules over the infinite rank Lie superalgebras $\widehat {\mathrm {gl}}_{\infty |\infty }$, $\widehat {\mathcal {C}}$ and $\widehat {\mathcal { D}}$, and determine the necessary and sufficient conditions for such modules to be unitarizable. The unitarizable irreducible modules are constructed in terms of Fock spaces of free quantum fields, and explicit formulae for their formal characters are also obtained by investigating Howe dualities between the infinite rank Lie superalgebras and classical Lie groups.

References

A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math. 64 (1987), no. 2, 118–175. MR 884183, DOI 10.1016/0001-8708(87)90007-7
Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1995. Translated from the German manuscript; Corrected reprint of the 1985 translation. MR 1410059
Shun-Jen Cheng and Ngau Lam, Infinite-dimensional Lie superalgebras and hook Schur functions, Comm. Math. Phys. 238 (2003), no. 1-2, 95–118. MR 1989670, DOI 10.1007/s00220-003-0819-3
Shun-Jen Cheng, Ngau Lam, and R. B. Zhang, Character formula for infinite-dimensional unitarizable modules of the general linear superalgebra, J. Algebra 273 (2004), no. 2, 780–805. MR 2037723, DOI 10.1016/S0021-8693(03)00538-6
Shun-Jen Cheng and Weiqiang Wang, Howe duality for Lie superalgebras, Compositio Math. 128 (2001), no. 1, 55–94. MR 1847665, DOI 10.1023/A:1017594504827
Shun-Jen Cheng and Weiqiang Wang, Remarks on the Schur-Howe-Sergeev duality, Lett. Math. Phys. 52 (2000), no. 2, 143–153. MR 1786858, DOI 10.1023/A:1007668930652
Shun-Jen Cheng and Weiqiang Wang, Lie subalgebras of differential operators on the super circle, Publ. Res. Inst. Math. Sci. 39 (2003), no. 3, 545–600. MR 2001187
Shun-Jen Cheng and R. B. Zhang, Howe duality and combinatorial character formula for orthosymplectic Lie superalgebras, Adv. Math. 182 (2004), no. 1, 124–172. MR 2028498, DOI 10.1016/S0001-8708(03)00076-8
Edward Frenkel, Victor Kac, Andrey Radul, and Weiqiang Wang, $\scr W_{1+\infty }$ and $\scr W(\mathfrak {g}\mathfrak {l}_N)$ with central charge $N$, Comm. Math. Phys. 170 (1995), no. 2, 337–357. MR 1334399
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR 1606831
Koji Hasegawa, Spin module versions of Weyl’s reciprocity theorem for classical Kac-Moody Lie algebras—an application to branching rule duality, Publ. Res. Inst. Math. Sci. 25 (1989), no. 5, 741–828. MR 1031225, DOI 10.2977/prims/1195172705
Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570. MR 986027, DOI 10.1090/S0002-9947-1989-0986027-X
Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR 1321638
V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Victor Kac and Andrey Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Comm. Math. Phys. 157 (1993), no. 3, 429–457. MR 1243706
Victor G. Kac, Weiqiang Wang, and Catherine H. Yan, Quasifinite representations of classical Lie subalgebras of $\scr W_{1+\infty }$, Adv. Math. 139 (1998), no. 1, 56–140. MR 1652526, DOI 10.1006/aima.1998.1753
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Maxim Nazarov, Capelli identities for Lie superalgebras, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 6, 847–872 (English, with English and French summaries). MR 1476298, DOI 10.1016/S0012-9593(97)89941-7
G. I. Ol′shanskiĭ and M. C. Prati, Extremal weights of finite-dimensional representations of the Lie superalgebra ${\mathfrak {g}}{\mathfrak {l}}_{n|m}$, Nuovo Cimento A (11) 85 (1985), no. 1, 1–18 (English, with Italian and Russian summaries). MR 789738, DOI 10.1007/BF02902385
Alexander Sergeev, An analog of the classical invariant theory for Lie superalgebras. I, II, Michigan Math. J. 49 (2001), no. 1, 113–146, 147–168. MR 1827078, DOI 10.1307/mmj/1008719038
Alexander Sergeev, An analog of the classical invariant theory for Lie superalgebras. I, II, Michigan Math. J. 49 (2001), no. 1, 113–146, 147–168. MR 1827078, DOI 10.1307/mmj/1008719038
Weiqiang Wang, Duality in infinite-dimensional Fock representations, Commun. Contemp. Math. 1 (1999), no. 2, 155–199. MR 1696098, DOI 10.1142/S0219199799000080
Weiqiang Wang, Dual pairs and infinite dimensional Lie algebras, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998) Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 453–469. MR 1745273, DOI 10.1090/conm/248/03836