On principal realization of modules for the affine Lie algebra 𝐴₁⁽¹⁾ at the critical level

Adamović, Dražen; Jing, Naihuan; Misra, Kailash

doi:10.1090/tran/7009

On principal realization of modules for the affine Lie algebra $A_1^{(1)}$ at the critical level
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by Dražen Adamović, Naihuan Jing and Kailash C. Misra PDF

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Abstract:

We present complete realization of irreducible $A_1^{(1)}$–modules at the critical level in the principal gradation. Our construction uses vertex algebraic techniques, the theory of twisted modules and representations of Lie conformal superalgebras. We also provide an alternative Z-algebra approach to this construction. All irreducible highest weight $A_1^{(1)}$–modules at the critical level are realized on the vector space $M_{\tfrac {1}{2}+\mathbb {Z}}(1)^{\otimes 2}$ where $M_{\tfrac {1}{2} + \mathbb {Z}}(1)$ is the polynomial ring ${\mathbb {C}}[\alpha (-1/2), \alpha (-3/2),\dots ]$. Explicit combinatorial bases for these modules are also given.

References

Dražen Adamović, Representations of the $N=2$ superconformal vertex algebra, Internat. Math. Res. Notices 2 (1999), 61–79. MR 1670180, DOI 10.1155/S1073792899000033
Dražen Adamović, Lie superalgebras and irreducibility of $A^{(1)}_1$-modules at the critical level, Comm. Math. Phys. 270 (2007), no. 1, 141–161. MR 2276443, DOI 10.1007/s00220-006-0153-7
Dražen Adamović, A classification of irreducible Wakimoto modules for the affine Lie algebra $A^{(1)}_1$, Recent advances in representation theory, quantum groups, algebraic geometry, and related topics, Contemp. Math., vol. 623, Amer. Math. Soc., Providence, RI, 2014, pp. 1–12. MR 3288618, DOI 10.1090/conm/623/12454
Dražen Adamović and Antun Milas, The $N=1$ triplet vertex operator superalgebras: twisted sector, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 087, 24. MR 2470509, DOI 10.3842/SIGMA.2008.087
Tomoyuki Arakawa and Peter Fiebig, The linkage principle for restricted critical level representations of affine Kac-Moody algebras, Compos. Math. 148 (2012), no. 6, 1787–1810. MR 2999304, DOI 10.1112/S0010437X12000395
K. Barron, Twisted modules for N=2 supersymmetric vertex operator superalgebras arising from finite automorphisms of the N=2 Neveu-Schwarz algebra, arXiv:1110.0229.
Chongying Dong and James Lepowsky, Generalized vertex algebras and relative vertex operators, Progress in Mathematics, vol. 112, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1233387, DOI 10.1007/978-1-4612-0353-7
Chongying Dong and Geoffrey Mason, On quantum Galois theory, Duke Math. J. 86 (1997), no. 2, 305–321. MR 1430435, DOI 10.1215/S0012-7094-97-08609-9
Jonathan Dunbar, Naihuan Jing, and Kailash C. Misra, Realization of $\widehat {\mathfrak {sl}}_2(\Bbb {C})$ at the critical level, Commun. Contemp. Math. 16 (2014), no. 2, 1450006, 13. MR 3195157, DOI 10.1142/S0219199714500060
Yuji Hara, Michio Jimbo, Hitoshi Konno, Satoru Odake, and Jun’ichi Shiraishi, On Lepowsky-Wilson’s $\scr Z$-algebra, Recent developments in infinite-dimensional Lie algebras and conformal field theory (Charlottesville, VA, 2000) Contemp. Math., vol. 297, Amer. Math. Soc., Providence, RI, 2002, pp. 143–149. MR 1919816, DOI 10.1090/conm/297/05096
Yuji Hara, Naihuan Jing, and Kailash Misra, BRST resolution for the principally graded Wakimoto module of $\widehat {\mathfrak {s}\mathfrak {l}}_2$, Lett. Math. Phys. 58 (2001), no. 3, 181–188 (2002). MR 1892918, DOI 10.1023/A:1014559525117
Boris L. Feigin and Edward V. Frenkel, Representations of affine Kac-Moody algebras and bosonization, Physics and mathematics of strings, World Sci. Publ., Teaneck, NJ, 1990, pp. 271–316. MR 1104262
Boris L. Feĭgin and Edward V. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys. 128 (1990), no. 1, 161–189. MR 1042449, DOI 10.1007/BF02097051
Edward Frenkel, Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005), no. 2, 297–404. MR 2146349, DOI 10.1016/j.aim.2004.08.002
Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2nd ed., Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2004. MR 2082709, DOI 10.1090/surv/088
Edward Frenkel and Dennis Gaitsgory, Geometric realizations of Wakimoto modules at the critical level, Duke Math. J. 143 (2008), no. 1, 117–203. MR 2414746, DOI 10.1215/00127094-2008-017
Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494, viii+64. MR 1142494, DOI 10.1090/memo/0494
Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026
Yongcun Gao and Haisheng Li, Generalized vertex algebras generated by parafermion-like vertex operators, J. Algebra 240 (2001), no. 2, 771–807. MR 1841356, DOI 10.1006/jabr.2001.8754
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Victor Kac, Vertex algebras for beginners, 2nd ed., University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1998. MR 1651389, DOI 10.1090/ulect/010
V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite-dimensional Lie algebras, Adv. in Math. 34 (1979), no. 1, 97–108. MR 547842, DOI 10.1016/0001-8708(79)90066-5
James Lepowsky and Robert Lee Wilson, Construction of the affine Lie algebra $A_{1}^{{}}(1)$, Comm. Math. Phys. 62 (1978), no. 1, 43–53. MR 573075, DOI 10.1007/BF01940329
James Lepowsky and Robert Lee Wilson, A new family of algebras underlying the Rogers-Ramanujan identities and generalizations, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 12, 7254–7258. MR 638674, DOI 10.1073/pnas.78.12.7254
James Lepowsky and Robert Lee Wilson, The structure of standard modules. I. Universal algebras and the Rogers-Ramanujan identities, Invent. Math. 77 (1984), no. 2, 199–290. MR 752821, DOI 10.1007/BF01388447
Hai-Sheng Li, Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, Moonshine, the Monster, and related topics (South Hadley, MA, 1994) Contemp. Math., vol. 193, Amer. Math. Soc., Providence, RI, 1996, pp. 203–236. MR 1372724, DOI 10.1090/conm/193/02373
Haisheng Li, On abelian coset generalized vertex algebras, Commun. Contemp. Math. 3 (2001), no. 2, 287–340. MR 1831932, DOI 10.1142/S0219199701000366
James Lepowsky and Haisheng Li, Introduction to vertex operator algebras and their representations, Progress in Mathematics, vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2023933, DOI 10.1007/978-0-8176-8186-9
Minoru Wakimoto, Fock representations of the affine Lie algebra $A^{(1)}_1$, Comm. Math. Phys. 104 (1986), no. 4, 605–609. MR 841673, DOI 10.1007/BF01211068
Xiaoping Xu, Introduction to vertex operator superalgebras and their modules, Mathematics and its Applications, vol. 456, Kluwer Academic Publishers, Dordrecht, 1998. MR 1656671, DOI 10.1007/978-94-015-9097-6