The Gelfand-Kirillov dimensions of algebras arising from representation theory (Q2759663)

The author studies graded algebras that arise from the induced representations for (i) semisimple algebraic groups and (ii) quantum groups. These algebras play an important role in the study of the cohomology groups of line bundles over flag varieties.NEWLINENEWLINENEWLINELet \(D\subset H\) be a subalgebra of a Hopf algebra \(H\) and \(M\) be a \(D\)-module. Under some assumptions, \(R=\text{Ind}_D^HM\) has the structure of an algebra. Let \(G\) be a simply connected semisimple algebraic group with Borel subgroup \(B\), in this case \(H\) is a distribution algebra of \(G\) and \(D\) is a distribution algebra of \(B\). In case of quantum groups, the author considers similar settings. The note concentrates on the calculation of the Gelfand-Kirillov dimension of these algebras \(R\).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00049].