Some remarks on Koszul algebras and quantum groups

Abstract

The q quantum groups of recent interest have noncommutative(if qeq 1) coordinate algebras which also have anoncocommutative coalgebra structure. The algebra structure isquadratic in nature. For example,M\sb q(2)=k[X\sb {11},X\sb {12},X\sb {21},X\sb {22}] has amongits relations X\sb {11}X\sb {12}=q\sp { 1}X\sb {12}X\sb {11}and X\sb {11}X\sb {22} X\sb {22}X\sb {11}=(q\sp { 1}q)X\sb {12}X\sb {21}. The coalgebra structure is that ofcomatrix units, e.g.,ΔX\sb {11}=\break X\sb {11}\otimes X\sb {11}+X\sb {12}\otimes X\sb {21}.The author first notes that M\sb q(2) is the quantum space oflinear endomorphisms of the quantum plane k[x,y] with relationxy=q\sp { 1}yx, and also of a q Grassmannian quantum plane.The main part of the paper studies quadratic algebras which aregraded by the nonnegative integers and are finitely generated andfinitely presented by elements of degree one with quadraticrelations. Three products of such algebras are given. One is theusual tensor product, and another is obtained by juxtaposingquadratic relations from each factor and then inverting thesecond and third letters in each resulting four letter word. Thislatter product has a corresponding internal Hom functor.Dualizing gives quantum semigroups of coendomorphisms ofquadratic algebras and certain comodules. At the same time, aKoszul complex is associated with each morphism of quadraticalgebras. Several constructions of quantum determinants aregiven. Finally, the previous constructions are carried over intothe Yang Baxter tensor categories of linearspaces.\par {Reviewer's remark: The notation in this reviewdiffers from that of the article under review. The reviewer feelsthat his notation avoids confusing functions with the objects onwhich they operate.}