Irreducible finite-dimensional representations of equivariant map algebras
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- by Erhard Neher, Alistair Savage and Prasad Senesi PDF
- Trans. Amer. Math. Soc. 364 (2012), 2619-2646
Abstract:
Suppose a finite group acts on a scheme $X$ and a finite-dimensional Lie algebra $\mathfrak g$. The corresponding equivariant map algebra is the Lie algebra $\mathfrak M$ of equivariant regular maps from $X$ to $\mathfrak g$. We classify the irreducible finite-dimensional representations of these algebras. In particular, we show that all such representations are tensor products of evaluation representations and one-dimensional representations, and we establish conditions ensuring that they are all evaluation representations. For example, this is always the case if $\mathfrak M$ is perfect.
Our results can be applied to multiloop algebras, current algebras, the Onsager algebra, and the tetrahedron algebra. Doing so, we easily recover the known classifications of irreducible finite-dimensional representations of these algebras. Moreover, we obtain previously unknown classifications of irreducible finite-dimensional representations of other types of equivariant map algebras, such as the generalized Onsager algebra.
References
- Bruce Allison, Stephen Berman, John Faulkner, and Arturo Pianzola, Realization of graded-simple algebras as loop algebras, Forum Math. 20 (2008), no. 3, 395–432. MR 2418198, DOI 10.1515/FORUM.2008.020
- Bruce Allison, Stephen Berman, John Faulkner, and Arturo Pianzola, Multiloop realization of extended affine Lie algebras and Lie tori, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4807–4842. MR 2506428, DOI 10.1090/S0002-9947-09-04828-4
- Bruce Allison, Stephen Berman, and Arturo Pianzola, Iterated loop algebras, Pacific J. Math. 227 (2006), no. 1, 1–41. MR 2247871, DOI 10.2140/pjm.2006.227.1
- Punita Batra, Representations of twisted multi-loop Lie algebras, J. Algebra 272 (2004), no. 1, 404–416. MR 2029040, DOI 10.1016/j.jalgebra.2003.09.044
- N. Bourbaki, Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1261, Hermann, Paris, 1958 (French). MR 0098114
- N. Bourbaki, Éléments de mathématique. Fascicule XXVII. Algèbre commutative. Chapitre 1: Modules plats. Chapitre 2: Localisation, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1290, Hermann, Paris, 1961 (French). MR 0217051
- N. Bourbaki, Éléments de mathématique. Algèbre. Chapitres 1 à 3, Hermann, Paris, 1970 (French). MR 0274237
- N. Bourbaki, Éléments de mathématique. Fasc. XXVI. Groupes et algèbres de Lie. Chapitre I: Algèbres de Lie, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1285, Hermann, Paris, 1971 (French). Seconde édition. MR 0271276
- N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Hermann, Paris, 1975 (French). Actualités Sci. Indust., No. 1364. MR 0453824
- Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1985 (French). Algèbre commutative. Chapitres 5 à 7. [Commutative algebra. Chapters 5–7]; Reprint. MR 782297
- Vyjayanthi Chari, Integrable representations of affine Lie-algebras, Invent. Math. 85 (1986), no. 2, 317–335. MR 846931, DOI 10.1007/BF01389093
- Vyjayanthi Chari, Ghislain Fourier, and Tanusree Khandai, A categorical approach to Weyl modules, Transform. Groups 15 (2010), no. 3, 517–549. MR 2718936, DOI 10.1007/s00031-010-9090-9
- Vyjayanthi Chari, Ghislain Fourier, and Prasad Senesi, Weyl modules for the twisted loop algebras, J. Algebra 319 (2008), no. 12, 5016–5038. MR 2423816, DOI 10.1016/j.jalgebra.2008.02.030
- Vyjayanthi Chari and Jacob Greenstein, An application of free Lie algebras to polynomial current algebras and their representation theory, Infinite-dimensional aspects of representation theory and applications, Contemp. Math., vol. 392, Amer. Math. Soc., Providence, RI, 2005, pp. 15–31. MR 2189867, DOI 10.1090/conm/392/07350
- Vyjayanthi Chari and Adriano A. Moura, Spectral characters of finite-dimensional representations of affine algebras, J. Algebra 279 (2004), no. 2, 820–839. MR 2078944, DOI 10.1016/j.jalgebra.2004.01.015
- Vyjayanthi Chari and Andrew Pressley, New unitary representations of loop groups, Math. Ann. 275 (1986), no. 1, 87–104. MR 849057, DOI 10.1007/BF01458586
- 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 and Andrew Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191–223. MR 1850556, DOI 10.1090/S1088-4165-01-00115-7
- Etsuro Date and Shi-shyr Roan, The structure of quotients of the Onsager algebra by closed ideals, J. Phys. A 33 (2000), no. 16, 3275–3296. MR 1766989, DOI 10.1088/0305-4470/33/16/316
- David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. MR 1730819
- B. Feigin and S. Loktev, Multi-dimensional Weyl modules and symmetric functions, Comm. Math. Phys. 251 (2004), no. 3, 427–445. MR 2102326, DOI 10.1007/s00220-004-1166-8
- Brian Hartwig, The tetrahedron algebra and its finite-dimensional irreducible modules, Linear Algebra Appl. 422 (2007), no. 1, 219–235. MR 2299006, DOI 10.1016/j.laa.2006.09.024
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original. MR 1834454, DOI 10.1090/gsm/034
- Brian Hartwig and Paul Terwilliger, The tetrahedron algebra, the Onsager algebra, and the $\mathfrak {sl}_2$ loop algebra, J. Algebra 308 (2007), no. 2, 840–863. MR 2295093, DOI 10.1016/j.jalgebra.2006.09.011
- Nathan Jacobson, Basic algebra. II, 2nd ed., W. H. Freeman and Company, New York, 1989. MR 1009787
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- Michael Lau, Representations of multiloop algebras, Pacific J. Math. 245 (2010), no. 1, 167–184. MR 2602688, DOI 10.2140/pjm.2010.245.167
- Haisheng Li, On certain categories of modules for affine Lie algebras, Math. Z. 248 (2004), no. 3, 635–664. MR 2097377, DOI 10.1007/s00209-004-0674-8
- George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
- Erhard Neher, Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004), no. 3, 90–96 (English, with English and French summaries). MR 2083842
- S. Eswara Rao, On representations of loop algebras, Comm. Algebra 21 (1993), no. 6, 2131–2153. MR 1215561, DOI 10.1080/00927879308824668
- S. Eswara Rao, Classification of irreducible integrable modules for multi-loop algebras with finite-dimensional weight spaces, J. Algebra 246 (2001), no. 1, 215–225. MR 1872618, DOI 10.1006/jabr.2001.8967
- S.-s. Roan, Onsager’s algebra, loop algebra and chiral pots model, Max-Planck-Institut Preprint MPI/91-70, 1991.
- Prasad Senesi, The block decomposition of finite-dimensional representations of twisted loop algebras, Pacific J. Math. 244 (2010), no. 2, 335–357. MR 2587435, DOI 10.2140/pjm.2010.244.335
- Prasad Senesi, Finite-dimensional representation theory of loop algebras: a survey, Quantum affine algebras, extended affine Lie algebras, and their applications, Contemp. Math., vol. 506, Amer. Math. Soc., Providence, RI, 2010, pp. 263–283. MR 2642570, DOI 10.1090/conm/506/09944
- D. B. Uglov and I. T. Ivanov, $\textrm {sl}(N)$ Onsager’s algebra and integrability, J. Statist. Phys. 82 (1996), no. 1-2, 87–113. MR 1372652, DOI 10.1007/BF02189226
Additional Information
- Erhard Neher
- Affiliation: Department of Mathematics, University of Ottawa, Ottawa, Ontario, Canada
- Email: erhard.neher@uottawa.ca
- Alistair Savage
- Affiliation: Department of Mathematics, University of Ottawa, Ottawa, Ontario, Canada
- Email: alistair.savage@uottawa.ca
- Prasad Senesi
- Affiliation: Department of Mathematics, The Catholic University of America, Washington, DC 20016
- Email: senesi@cua.edu
- Received by editor(s): April 12, 2010
- Received by editor(s) in revised form: July 12, 2010
- Published electronically: December 29, 2011
- Additional Notes: The first and second authors gratefully acknowledge support from the NSERC through their respective Discovery grants.
The third author was partially supported by the Discovery grants of the first two authors. - © Copyright 2011 Erhard Neher, Alistair Savage, Prasad Senesi
- Journal: Trans. Amer. Math. Soc. 364 (2012), 2619-2646
- MSC (2010): Primary 17B10, 17B20, 17B65
- DOI: https://doi.org/10.1090/S0002-9947-2011-05420-6
- MathSciNet review: 2888222