Central extensions of generalized orthosymplectic Lie superalgebras

Abstract

The key ingredient of this paper is the universal central extension of the generalized orthosymplectic Lie superalgebra osp m|2n (R, − ) coordinatized by a unital associative superalgebra (R, − ) with superinvolution. Such a universal central extension will be constructed via a Steinberg orthosymplectic Lie superalgebra coordinated by (R, − ). The research on the universal central extension of osp m|2n (R, − ) will yield an identification between the second homology group of the generalized orthosymplectic Lie superalgebra osp m|2n (R, − ) and the first Z/2Z-graded skew-dihedral homology group of (R, − ) for (m, n) = (2, 1), (1, 1). The universal central extensions of osp 2|2 (R, − ) and osp 1|2 (R, − ) will also be treated separately.MSC(2010): 17B05, 19D55.

Remarks on the Second Homology Groups of Queer Lie Superalgebras

“…The universal central extensions of sl n (R), n 3 have been completely determined in [3], which has been further generalized to the Lie superalgebra sl m|n (R), m + n 3 in [4] and [7]. The authors addressed in [1] and [2] the case of an otho-symplectic Lie superalgebra and the case of a periplectic Lie superalgebra that are coordinatized by an associative superalgebra with a super anti-involution. These Lie superalgebras are Super analogy of unitary Lie algebra, thus it can help us to identify the second homology groups of these Lie superalgebras with the Z/2Z-graded version of the first dihedral and skew-dihedral homology group.…”

Second homology of generalized periplectic Lie superalgebras

“…The isomorphism between the second homology of the Lie superalgebra sl m|n (S) coordinated by a unital associative superalgebra S with m + n 5 and the first Z/2Z-graded cyclic homology HC 1 (S) was established. Recent investigation [2] further gave the identification between the second homology of the ortho-symplectic Lie superalgebra osp m|2n (R, − ) and the first Z/2Z-graded skew-dihedral homology of (R, − ) for (m, n) = (1, 1) or (2,1), where (R, − ) is a unital associative superalgebra with superinvolution (see (2.1) for the definition). A series of deep investigations on the relationship between the homology theory of Lie algebras and the homology theory of associative algebras have been made in [12,13].…”