Low Degree Morphisms of E (5, 10)-Generalized Verma Modules
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Low Degree Morphisms of E (5, 10)-Generalized Verma Modules Nicoletta Cantarini1 · Fabrizio Caselli1 Received: 28 March 2019 / Accepted: 16 September 2019 / © Springer Nature B.V. 2019
Abstract In this paper we face the study of the representations of the exceptional Lie superalgebra E(5, 10). We recall the construction of generalized Verma modules and give a combinatorial description of the restriction to sl5 of the Verma module induced by the trivial representation. We use this description to classify morphisms between Verma modules of degree one, two and three proving in these cases a conjecture given by Rudakov (2010). A key tool is the notion of dual morphism between Verma modules. Keywords Lie superalgebras · Verma modules · Singular vectors Mathematics Subject Classification (2010) Primary: 17B15 · 17B25; Secondary 17B65 · 17B70
1 Introduction Infinite dimensional linearly compact simple Lie superalgebras over the complex numbers were classified by Victor Kac in 1998 [3]. A complete list, up to isomorphisms, consists of ten infinite series and five exceptions, denoted by E(1, 6), E(3, 6), E(3, 8), E(5, 10) and E(4, 4). See also [1, 9–11] for the genesis of these superalgebras. Some years later Kac and Rudakov initiated the study of the representations of these algebras [4–7] developing a general theory of Verma modules that we briefly recall.
Presented by: Peter Littelmann Fabrizio Caselli
[email protected] Nicoletta Cantarini [email protected] 1
Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
N. Cantarini, F. Caselli
Let L = ⊕j ∈Z Lj be a Z-graded Lie superalgebra, let L− = ⊕j 0 Lj and L≥0 = L0 ⊕ L+ . We denote by U (L) the universal enveloping algebra of L. If F is an irreducible L0 -module we define M(F ) = U (L) ⊗U (L≥0 ) F where we extend the action of L0 to L≥0 by letting L+ act trivially on F . We call M(F ) a minimal generalized Verma module associated to F . If M(F ) is not irreducible we say that it is degenerate. In [4–7] a complete description of the degenerate Verma modules for E(3, 6) and E(3, 8) is given, as well as of their unique irreducible quotients. In [6] some basic ideas and constructions are settled also for E(5, 10). In this case Kac and Rudakov conjecture a complete list of L0 -modules which give rise to the degenerate Verma modules (see Conjecture 4.6). In 2010 Rudakov tackled the proof of the conjecture through the study of morphisms between Verma modules. The existence of a degenerate Verma module is indeed strictly related to the existence of such morphisms of positive degree (see Proposition 3.5). In [8] Rudakov classified morphisms of degree one and gave some examples of morphisms of degree at most 5. He also conjectured that there exists no morphism of higher degree and that his list exhausts all the examples. A more general family of modules, possibly induced from infinite-dimensional sl5 -modules, had been studied in [2], where some of Rudakov’s examples in degree one and two had been obtained
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