Central extensions of generalized orthosymplectic Lie superalgebras

We completely determine the universal central extension of the generalized orthosymplectic Lie superalgebra ospm 12n（R,-） that is coordinatized by an arbitrary unital associative superalgebra （R,-） with superinvolution. As a result, an identification between the second homology group of the Lie superalgebra ospm｜2n （R,-） and the first skew-dihedral homology group of the associative superalgebra （R,-） with superin-volution is created for positive integers m and n with （m, n）≠ （1, 1） and （m, n）≠（2, 1）. The second homology groups of the Lie superalgebras ospm1｜2（R,-） and ospm｜2n （R,-） are also characterized explicitly.

We completely determine the universal central extension of the generalized orthosymplectic Lie superalgebra ospm|2n(R,-)that is coordinatized by an arbitrary unital associative superalgebra(R,-)with superinvolution.As a result,an identification between the second homology group of the Lie superalgebra ospm|2n(R,-)and the first skew-dihedral homology group of the associative superalgebra(R,-)with superinvolution is created for positive integers m and n with(m,n)=(1,1)and(m,n)=(2,1).The second homology groups of the Lie superalgebras osp1|2(R,-)and osp2|2(R,-)are also characterized explicitly.