Combinatorial bases of modules for affine Lie algebra B 2(1)

Mirko Primc

Open Mathematics, Tome 11 (2013), p. 197-225 / Harvested from The Polish Digital Mathematics Library

Résumé

We construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases and the same presentation P/I.

Détail

Publié le : 2013-01-01

Zbl 1293.17032

EUDML-ID : urn:eudml:doc:268992

Bibliographie

[1] Ardonne E., Kedem R., Stone M., Fermionic characters and arbitrary highest-weight integrable $ \widehat{\mathfrak {s}\mathfrak {l}}_{r + 1} $ -modules, Comm. Math. Phys., 2006, 264(2), 427–464 http://dx.doi.org/10.1007/s00220-005-1486-3 | Zbl 1233.17017

[2] Baranović I., Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of level 2 standard modules for D 4(1), Comm. Algebra, 2011, 39(3), 1007–1051 http://dx.doi.org/10.1080/00927871003639329 | Zbl 1269.17011

[3] Calinescu C., Principal subspaces of higher-level standard $ \widehat{\mathfrak {s}\mathfrak {l}(3)} $ -modules, J. Pure Appl. Algebra, 2007, 210(2), 559–575 http://dx.doi.org/10.1016/j.jpaa.2006.10.018

[4] Calinescu C., Lepowsky J., Milas A., Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of types A,D,E, J. Algebra, 2010, 323(1), 167–192 http://dx.doi.org/10.1016/j.jalgebra.2009.09.029 | Zbl 1221.17032

[5] Capparelli S., Lepowsky J., Milas A., The Rogers-Ramanujan recursion and intertwining operators, Commun. Contemp. Math., 2003, 5(6), 947–966 http://dx.doi.org/10.1142/S0219199703001191 | Zbl 1039.05005

[6] Capparelli S., Lepowsky J., Milas A., The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators, Ramanujan J., 2006, 12(3), 379–397 http://dx.doi.org/10.1007/s11139-006-0150-7 | Zbl 1166.17009

[7] Dong C., Lepowsky J., Generalized vertex algebras and relative vertex operators, Progr. Math., 112, Birkhäuser, Boston, 1993 http://dx.doi.org/10.1007/978-1-4612-0353-7 | Zbl 0803.17009

[8] Dong C., Li H., Mason G., Simple currents and extensions of vertex operator algebras, Comm. Math. Phys., 1996, 180(3), 671–707 http://dx.doi.org/10.1007/BF02099628 | Zbl 0873.17027

[9] Feigin B., Jimbo M., Loktev S., Miwa T., Mukhin E., Bosonic formulas for (k, l)-admissible partitions, Ramanujan J., 2003, 7(4), 485–517; Addendum to “Bosonic formulas for (k, l)-admissible partitions”, Ramanujan J., 2003, 7(4), 519–530 http://dx.doi.org/10.1023/B:RAMA.0000012430.68976.c0 | Zbl 1039.05008

[10] Feigin B., Jimbo M., Miwa T., Mukhin E., Takeyama Y., Fermionic formulas for (k, 3)-admissible configurations, Publ. Res. Inst. Math. Sci., 2004, 40(1), 125–162 http://dx.doi.org/10.2977/prims/1145475968 | Zbl 1134.17311

[11] Feigin B., Kedem R., Loktev S., Miwa T., Mukhin E., Combinatorics of the $ \widehat{\mathfrak {s}\mathfrak {l}}_2 $ spaces of coinvariants, Transform. Groups, 2001, 6(1), 25–52 http://dx.doi.org/10.1007/BF01236061 | Zbl 1004.17001

[12] Feigin E., The PBW filtration, Represent. Theory, 2009, 13, 165–181 http://dx.doi.org/10.1090/S1088-4165-09-00349-5 | Zbl 1229.17026

[13] Frenkel I.B., Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory, J. Funct. Anal., 1981, 44(3), 259–327 http://dx.doi.org/10.1016/0022-1236(81)90012-4

[14] Frenkel I.B., Huang Y.-Z., Lepowsky J., On Axiomatic Approaches to Vertex Operator Algebras and Modules, Mem. Amer. Math. Soc., 1993, 104, #494

[15] Frenkel I.B., Kac V.G., Basic representations of affine Lie algebras and dual resonance models, Invent. Math., 1980, 62(1), 23–66 http://dx.doi.org/10.1007/BF01391662 | Zbl 0493.17010

[16] Fuchs J., Simple WZW currents, Comm. Math. Phys., 1991, 136(2), 345–356 http://dx.doi.org/10.1007/BF02100029 | Zbl 0736.17033

[17] Georgiev G., Combinatorial constructions of modules for infinite-dimensional Lie algebras. I. Principal subspace, J. Pure Appl. Algebra, 1996, 112(3), 247–286 http://dx.doi.org/10.1016/0022-4049(95)00143-3

[18] Huang Y.-Z., Lepowsky J., Toward a theory of tensor products for representations of a vertex operator algebra, In: Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, New York, June 3–7, 1991, World Scientific, River Edge, 1992, 344–354 | Zbl 0829.17025

[19] Kac V.G., Infinite-Dimensional Lie Algebras, 3rd ed, Cambridge University Press, Cambridge, 1990 http://dx.doi.org/10.1017/CBO9780511626234 | Zbl 0716.17022

[20] Kang S.-J., Kashiwara M., Misra K.C., Miwa T., Nakashima T., Nakayashiki A., Affine crystals and vertex models, In: Infinite Analysis, Kyoto, June 1–August 31, 1991, Adv. Ser. Math. Phys., 16, World Scientific, River Edge, 1992, 449–484 | Zbl 0925.17005

[21] Lepowsky J., Primc M., Structure of the Standard Modules for the Affine Lie Algebra A 1(1), Contemp. Math., 46, American Mathematical Society, Providence, 1985 http://dx.doi.org/10.1090/conm/046 | Zbl 0569.17007

[22] Li H., Extension of vertex operator algebras by a self-dual simple module, J. Algebra, 1997, 187(1), 236–267 http://dx.doi.org/10.1006/jabr.1997.6795

[23] Li H., The physics superselection principle in vertex operator algebra theory, J. Algebra, 1997, 196(2), 436–457 http://dx.doi.org/10.1006/jabr.1997.7126

[24] Meurman A., Primc M., Vertex operator algebras and representations of affine Lie algebras, Acta Appl. Math., 1996, 44(1–2), 207–215 http://dx.doi.org/10.1007/BF00116522 | Zbl 0858.17024

[25] Meurman A., Primc M., Annihilating Fields of Standard Modules of sl(2,ℂ)∼ and Combinatorial Identities, Mem. Amer. Math. Soc., 1999, 137, #652 | Zbl 0918.17018

[26] Primc M., Vertex operator construction of standard modules for A n(1), Pacific J. Math., 1994, 162(1), 143–187 | Zbl 0787.17024

[27] Primc M., Basic representations for classical affine Lie algebras, J. Algebra, 2000, 228(1), 1–50 http://dx.doi.org/10.1006/jabr.1999.7899

[28] Primc M., (k, r)-admissible configurations and intertwining operators, In: Lie Algebras, Vertex Operator Algebras and Their Applications, Raleigh, May 17–21, 2005, Contemp. Math., 442, American Mathematical Society, Providence, 2007, 425–434 http://dx.doi.org/10.1090/conm/442/08540

[29] Stoyanovski A.V., Feigin B.L., Functional models of the representations of current algebras, and semi-infinite Schubert cells, Funct. Anal. Appl., 1994, 28(1), 55–72 http://dx.doi.org/10.1007/BF01079010 | Zbl 0905.17030

[30] Trupčević G., Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of higher-level standard $ \widetilde{\mathfrak {s}\mathfrak {l}}(\ell + 1,\mathbb {C}) $ -modules, J. Algebra, 2009, 322(10), 3744–3774 http://dx.doi.org/10.1016/j.jalgebra.2009.07.024 | Zbl 1216.17008