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 \( {\frak g}_k \otimes_k A \), regarded as a Lie superalgebra over k, where \( {\frak g}_k \) 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.