On capability and the Schur multiplier of some nilpotent Lie superalgebras

A Lie superalgebra H satisfying H2=Z(H) is called generalized Heisenberg Lie superalgebra. If d(H):=(m∣n) is the minimum number of generators required to describe H, then in this article we intend to find the structure of H when precisely dim⁡(H2)=12(m+n)2+(n−m). Further, we give some results about the capability, and the Schur multiplier of H. Moreover, we find multiplier M(L) for any nilpotent Lie superalgebra L of nilpotency class 2 with d(L)=(r∣s) when its derived subalgebra is of maximum possible dimension and finally show that such an algebra is capable.