Low-dimensional cohomology of Lie superalgebra slm|n in with coefficients in Witt or Special superalgebras

Over a field of characteristic p > 2, the Witt superalgebras are viewed as modules of the special linear Lie superalgebra sl(m vertical bar n) by means of the adjoint representation. The low-dimensional cohomology groups of Elm In with coefficients in Witt superalgebras are computed by virtue of the direct sum decomposition of submodules and the weight space decompositions of these submodules relative to the standard Cartan subalgebra of sl(m vertical bar n). Then we are able to obtain the low-dimensional cohomology groups of sl(m vertical bar n) with coefficients in Special superalgebras, which are sl(m vertical bar n)-submodules of Witt superalgebras. (C) 2013 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.