In this paper, first we prove that all finite dimensional special Heisenberg Lie superalgebras with even center have same dimension, say (2m + 1 | n) for some non-negative integers m, n and are isomorphism with them. Further, for a nilpotent Lie superalgebra, where M(L) denotes the Schur multiplier of L. Moreover, if (r, s) = (1, 0) (respectively (r, s) = (0, 1)), then the equality holds if and only if L ∼ = H(1, 0) ⊕ A1 (respectively H(0, 1) ⊕ A2), where A1 and A2 are abelian Lie superalgebras with dim and H(1, 0), H(0, 1) are special Heisenberg Lie superalgebras of dimension 3 and 2 respectively.
In this paper using the concept of isoclinism, we give the structure of all covers of Lie superalgebras when their Schur multipliers are finite dimensional. Further it has been shown that, each stem extension of a finite dimensional Lie superalgebra is a homomorphic image of a stem cover for it and as a corollary it is concluded that maximal stem extensions of Lie superalgebras are precisely same as the stem covers. Moreover, we have defined stem Lie superalgebra and show that a Lie superalgebra with finite dimensional derived subalgebra and finitely generated central factor is isoclinic to a finite dimensional Lie superalgebra.
We show that HD(4, 1) hyperbolic Kac-Moody superalgebra of rank 6 contains every simply laced Kac-Moody superalgebra with degenerate odd root as a Lie subalgebra. Our result is the supersymmetric extension of earlier work [S. Viswanath, “Embeddings of HyperbolicKac-Moody Algebras into E10,” Lett. Math. Phys. 83, 139–148 (2008)]10.1007/s11005-007-0214-7 for hyperbolic Kac-Moody algebra.
Isoclinism of Lie superalgebras has been defined and studied currently. In this article it is shown that for finite dimensional Lie superalgebras of same dimension, the notation of isoclinism and isomorphism are equivalent. Furthermore we show that covers of finite dimensional Lie superalgebras are isomorphic using isoclinism concept.2010 Mathematics Subject Classification. Primary 17B30; Secondary 17B05.
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