Generalized P(N)-graded Lie superalgebras

Abstract

We generalize the P(N)-graded Lie superalgebras of Martinez-Zelmanov. This generalization is not so restrictive but suffcient enough so that we are able to have a classification for this generalized P(N)-graded Lie superalgebras. Our result is that the generalized P(N)-graded Lie super-algebra L is centrally isogenous to a matrix Lie superalgebra coordinated by an associative superalgebra with a super-involution. Moreover, L is P(N)-graded if and only if the coordinate algebra R is commutative and the super-involution is trivial. This recovers Martinez-Zelmanov's theorem for type P(N). We also obtain a generalization of Kac's coordinatization via Tits-Kantor-Koecher construction. Actually, the motivation of this generalization comes from the Fermionic-Bosonic module construction.

Keywords

Root system graded Lie superalgebras / associative superalgebra / quantum tori / representations

References

[1] Benkart G, Elduque A. Lie superalgebras graded by the root system A(m, n). J Lie Theory, 2003, 13(2): 387–400

[2] Benkart G, Elduque A. Lie superalgebras graded by the root system B(m, n). Selecta Math (N S), 2003, 9: 313–360

[3] Benkart G, Elduque A, Martinez C. A(n, n)-graded Lie superalgebras. J Reine Angew Math, 2004, 573: 139–156

[4] Benkart G, Zelmanov E. Lie algebras graded by finite root systems and intersection matrix algebras. Invent Math, 1996, 126(1): 1–45

[5] Berman S, Gao Y, Krylyuk Y. Quantum tori and the structure of elliptic quasi-simple Lie algebras. J Funct Anal, 1996, 135(2): 339–389

[6] Berman S, Moody R V. Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy. Invent Math, 1992, 108(2): 323–347

[7] Chen H J, Gao Y. BC_N-graded Lie algebras arising from fermionic representations. J Algebra, 2007, 308(2): 545–566

[8] Cheng J, Zeng Z T. Irreducible Wakimoto-like Modules for the Lie superalgebra D(2,1;α). Acta Math Sin (Engl Ser), 2018, 40(34): 1578–1588

[9] Feingold A J, Frenkel I B. Classical affine algebras. Adv Math, 1985, 56(2): 117–172

[10] Frenkel I B. Spinor representations of affine Lie algebras. Proc Natl Acad Sci USA, 1980, 77(11): 6303–6306

[11] Gao Y. Fermionic and bosonic representations of the extended affine Lie algebragl glN(ℂq) ˜. Canad Math Bull, 2002, 45(4): 623–633

[12] Kac V G. Lie superalgebras. Adv Math, 1977, 26(1): 8–96

[13] Kac V G. Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras. Comm Algebra, 1977, 5(13): 1375–1400

[14] Kac V G, Peterson D H. Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc Natl Acad Sci USA, 1981, 78(6): 3308–3312

[15] Lau M. Bosonic and fermionic representations of Lie algebra central extensions. Adv Math, 2005, 194(2): 225–245

[16] Manin Yu I. Topics in Noncommutative Geometry. M B Porter Lecture Ser. Princeton: Princeton Univ Press, 1991

[17] Martinez C. Simplicity of Jordan superalgebras and relations with Lie structures. Irish Math Soc Bull, 2003, 50: 97–116

[18] Martinez C, Shestakov I, Zelmanov E. Jordan bimodules over the superalgebras P(n) and Q(n). Trans Amer Math Soc, 2010, 362(4): 2037–2051

[19] Martinez C, Zelmanov E. Lie superalgebras graded by P(n) and Q(n). Proc Natl Acad Sci USA, 2003, 100(14): 8130–8137

[20] Neher E. An introduction to universal central extensions of Lie superalgebras. In: Bahturin Yu, ed. Groups, Rings, Lie and Hopf Algebras. Math Appl, Vol 555. Berlin: Springer, 2003, 141–166

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