Central extensions of Lie superalgebras

For a commutative algebra A over a commutative ring k satisfying certain conditions, we construct the universal central extension of g k k A , regarded as a Lie superalgebra over k , where g k denotes a basic classical Lie superalgebra over k . To consider basic classical Lie superalgebras over an ring k , we also show the existence of their Chevalley basis. Our results contain not only the descriptions of the untwisted affine Lie superalgebras but also those of the toroidal Lie superalgebras.