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Finite-dimensional Hamiltonian Lie superalgebra. (English) Zbl 1021.17017

Let \(H=H(m,n,\underline{t})\) be the (finite-dimensional) Hamiltonian Lie superalgebra over an algebraically closed field \(F\) of characteristic \(p>3\). Generalizing the mixed product of [the reviewer, Sci. Sin., Ser. A 29, 570-581 (1986; Zbl 0601.17013)], the authors prove that if \(V\) is an irreducible module of the zero graded term \(P\) (which is an ortho-symplectic Lie superalgebra) with highest weight \(\lambda \) and \(\lambda \) is not a fundamental weight, then the mixed product \( \widetilde{V}(\lambda)\) is an irreducible module of \(H\). They also determine the (\({\mathbb Z}_2\)-homogeneous) superderivations \(D\) of \(H\) such that \(\dim \operatorname {Im}(D)\) are minimal. It follows that the natural filtration of \(H\) is invariant, and that \(H(m,n,\underline{t})\) and \(H(m^{\prime },n^{\prime },\underline{t}^{\prime })\) are isomorphic if and only if \( m=m^{\prime }\), \(n=n^{\prime }\) and \(\{\{t_1,t_{1^{\prime }}\},\cdots,\{t_r,t_{r^{\prime }}\}\}=\{\{t_1^{\prime },t_{1^{\prime }}^{\prime }\},\dots,\{t_r^{\prime },t_{r^{\prime }}^{\prime }\}\}.\)
The discussion is similar to that of the reviewer’s paper [Chin. Ann. Math. 2, 105-115 (1981; Zbl 0498.17009)].

MSC:

17B70 Graded Lie (super)algebras
17B50 Modular Lie (super)algebras
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