Representations of a central extension of the simple Lie superalgebra \(\mathfrak{p}(3)\)
From the introduction: ``Finite-dimensional simple Lie superalgebras over \(\mathbb C\) have been classified in [\textit{V. G. Kac}, Adv. Math. 26, 8--96 (1977; Zbl 0366.17012)]. Some of these superalgebras have non-trivial central extensions: \(\mathfrak{sl}(n|n)\) is a central extension of \(\mathfrak{psl}(n|n)\), \(\mathfrak{sq}(n)\) is a central extension of \(\mathfrak{psq}(n)\), the Poisson Lie superalgebra \(\mathcal{SP}(n)\) is a central extension of the special Hamiltonian superalgebra \(\mathcal{SH}(n)\), and finally the Lie superalgebra \(\hat{\mathfrak{p}}(3)\) is the unique (up to isomorphism) non-trivial central extension of the Lie superalgebra \(\mathfrak{p}(3)\). It is natural to study finite-dimensional representations of these central extensions as well as representations of the corresponding simple superalgebras. For \(\mathfrak{sl}(n|n)\) and \(\mathfrak{sq}(n)\) it is convenient to study representations of their respective Lie algebras of derivations \(\mathfrak{gl}(n|n)\) and \(\mathfrak{q}(n)\). Finite-dimensional representations of these superalgebras are studied in detail by several authors [see Refs. 2--4,10,11]. For the Lie superalgebra \(\mathfrak{psl}(2|2)\) there is a three-dimensional space of central extensions, irreducible finite-dimensional representations over those central extensions are studied in [\textit{T. Matsumoto} and \textit{A. Molev}, J. Math. Phys. 55, No. 9, 091704, 22 p. (2014; Zbl 1304.82015)]. Irreducible finite-dimensional representations of the Poisson superalgebra are described in [\textit{A. V. Shapovalov}, Serdica 7, 337--342 (1981; Zbl 0502.58016)]. As far as we know, the representations of \(\hat{\mathfrak{p}}(3)\) with nonzero central charge have not been previously studied. In the present paper we try to fill this gap: we compute characters of irreducible finite-dimensional irreducible representations of \(\hat{\mathfrak{p}}(3)\) with nonzero central charge. The case of zero central charge follows from more general results for the Lie superalgebra \(\mathfrak{p}(n)\) for arbitrary \(n\) in [\textit{M. Balagovic} et al., Math. Res. Lett. 26, No. 3, 643--710 (2019; Zbl 1435.17011)]. We plan to use the results of the present paper to compute the Cartan matrix of the category of finite-dimensional representations of \(\hat{\mathfrak{p}}(3)\) and classify the blocks of this category.''
scientific article; zbMATH DE number 6992468
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