LOW-DIMENSIONAL COHOMOLOGY OF LIE SUPERALGEBRA $A(1, 0)$ WITH COEFFICIENTS IN WITT OR SPECIAL SUPERALGEBRAS

Abstract

Over a field of characteristic $p \gt 2$; the low-dimensional cohomology groups of the special linear Lie superalgebra $\mathbf{A}(1;0)$ with coefficients in Witt superalgebra are computed by means of a direct sum decomposition of submodules and the weight space decomposition of the Witt superalgebra viewed as $\mathbf{A}(1;0)$-module. As application, the lowdimensional cohomology groups of $\mathbf{A}(1;0)$ with coefficients in Special superalgebra are also determined.