The general linear supergroup and its Hopf superalgebra of regular functions.
Let \({\mathfrak g}\) be the complex Lie superalgebra \({\mathfrak g}{\mathfrak l}(m, n)\). Using the continuous dual \(U({\mathfrak g})^0\) of the universal enveloping algebra of \({\mathfrak g}\), the authors construct and study the Hopf superalgebra \({\mathfrak B}\) of regular functions on the general linear supergroup \(G\). \(G\) can be recovered from \({\mathfrak B}\) via a Tannaka duality procedure. A Borel-Weil realization is obtained for the irreducible integral representations. Schur superalgebras are introduced and shown to be semisimple, and the simple ideals are explicitly determined. The main viewpoint is Hopf algebraic, which, the authors hope will be helpful in studying quantum supergroups. The authors' approach is related to that of \textit{H. Boseck} [ibid. 123, 61--72 (1985; Zbl 0579.17006), Math. Nach. 130, 137--138 (1987; Zbl 0613.17003), ibid. 143, 303--327 (1989; Zbl 0683.22014), ibid. 148, 81--115 (1990; Zbl 0721.22007)] and there is a small overlap in results, pointed out in the text.
scientific article; zbMATH DE number 1867969
scientific article