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I know sl(m|n) is the A-series Lie superalgebra by Kac

But in some literature, people also use the terminology su(m|n), which seems to be subclass of A-series. I am wondering how these two are related in general? For example, representeation theory or Lie algebra commutators?

Especially, people who study WZNW model on su(m|n) always start by discussing the sl(m|n) representations. I don't understand why. For example: https://arxiv.org/pdf/hep-th/0611147.pdf

jtkw
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    On Page $3$ it is said "The aim of this section is to decompose the space of functions on the supergroup $SU(2|1)$ into (generalized) eigenfunctions of the quadratic Casimir element in the regular representations. Since the Casimir commutes with the generator, the eigenspaces may be decomposed into representation of the Lie superalgebra $sl(2|1)$. It is therefore useful to have some background on the representation theory of $sl(2|1)$". – Dietrich Burde Feb 17 '21 at 12:37

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