Serre-type relations for special linear Lie superalgebras
A (complex) simple Lie algebra of rank \(r\) can be defined in terms of \(3r\) (Chevalley) generators and a number of relations (the Serre relations) in which the elements of the Cartan matrix \(A\) appear. Here, the analogous relations are considered for the case where \(A\) is the simplest Cartan matrix of the Lie superalgebra \(sl(m/n)\). Surprisingly, for \(m,n\geq 2\), the Lie superalgebra defined by these generators and relations is not \(sl(m/n)\) but a larger algebra, \(s(m/n)\). It is shown that \(sl(m/n)\) is a nontrivial quotient algebra of \(s(m/n)\), and the supplementary relations are presented. Following this, the relations previously used to define the \(q\)-deformed enveloping algebra \(U_ q(sl(m/n))\) should be modified accordingly.
scientific article; zbMATH DE number 89328