Abstract
The purpose of this paper is to investigate central elements in distribution algebras D i s t(G) of general linear supergroups G = G L(m|n). As an application, we compute explicitly the center of D i s t(G L(1|1)) and its image under Harish-Chandra homomorphism.
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References
Berezin, F. A.: Introduction to superanalysis. D. Reidel Publishing Co., Dordrecht (1987). expanded translation from Russian: V.P. Palamodov (Ed.), Introduction to algebra and analysis with anticommuting variables, Moscow State University, Moscow, 1983
Brundan, J., Kujawa, J.: A new proof of the Mullineux conjecture. J. Algebraic Combin. 18, 13–39 (2003)
Carter, R.W., Lustig, G.: On the modular representations of the general linear and symmetric groups. Math. Z. 136, 193–242 (1974)
Harish-Chandra: On some applications of the universal enveloping algebra of a semisimple Lie algebra. T.A.M.S. 70, 28–96 (1951)
Howe, R. E.: Harish-Chandra homomorphisms. The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998), pp 321–332. Proc. Sympos. Pure Math., 68, Amer. Math. Soc., Providence (2000)
Haboush, W.J.: Central differential operators on split semisimple groups over fields of positive characteristic. Lecture Notes in Math. 795, 35–85 (1980)
Grishkov, A.N., Zubkov, A.N.: Solvable, reductive and quasireductive supergroups. accepted in J.Algebra; see also arXiv:math.RT/1302.5648
Jacobson, N.: Structure of rings, vol. 37. American mathematical society, Colloquium Publications, Providence (1956)
Jantzen, J.: Representations of algebraic groups. Academic Press, New York (1987)
Kujawa, J.: The Steinberg tensor product theorem for G L(m|n). Representations of algebraic groups, quantum groups, and Lie algebras, pp 123–132. Contemp. Math., 413, Amer. Math. Soc., Providence (2006)
Kujawa, J.: Crystal structures arising from representations of G L(m|n). Represent. Theory 10, 49–85 (2006)
Marko, F., Zubkov, A.N.: Schur superalgebras in characteristic p. Algebr. Represent. Theory 9(1), 1–12 (2006)
Marko, F., Zubkov, A.N.: Pseudocompact algebras and highest weight categories. Algebr. Represent. Theory 16(3), 689–728 (2013)
Masuoka, A: The fundamental correspondence in super affine groups and super formal groups. J. Pure Appl. Algebra 202, 284–312 (2005)
Masuoka, A.: Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field. Transform. Groups 17(4), 1085–1121 (2012). (see also arXiv:1111.2387)
Masuoka, A., Zubkov, A.N.: Quotient sheaves of algebraic supergroups are superschemes. J. Algebra 348, 135–170 (2011)
Sweedler, M.E.: Hopf algebras. Benjamin, New York (1969)
Shu, B., Wang, W.: Modular representations of the ortho-symplectic supergroups. Proc. Lond. Math. Soc. (3) 96(1), 251–271 (2008)
Zubkov, A. N.: Affine quotients of supergroups. Transform. Groups 14(3), 713–745 (2009)
Zubkov, A.N.: Some properties of general linear supergroups and of Schur superalgebras. Algebra Logic 45(3), 147–171 (2006)
Zubkov, A.N.: Some homological properties of G L(m|n) in arbitrary characteristic. To appear in J. Algebra Appl., 10.1142/S021949881650119X (see also arXiv:1405.3890)
Zubkov, A.N.: On quotients of affine superschemes over finite supergroups. J. Algebra Appl. 10(3), 391–408 (2011)
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Presented by Vyjayanthi Chari.
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Zubkov, A.N., Marko, F. The Center of Dist (GL(m|n)) in Positive Characteristic. Algebr Represent Theor 19, 613–639 (2016). https://doi.org/10.1007/s10468-015-9591-2
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DOI: https://doi.org/10.1007/s10468-015-9591-2