Integration on Lie supergroups: A Hopf superalgebra approach

Let g=g0¯⊕g1¯ be a finite-dimensional complex Lie superalgebra such that g0¯ is reductive and the adjoint representation of g0¯ in g1¯ is completely reducible. A Lie supergroup associated to g is defined by fixing the Hopf superalgebra of regular functions on the supergroup. Then it is shown that on this Hopf superalgebra there exists a non-zero left integral. According to an earlier work by the authors, this integral is unique up to scalar multiples.