This is the first paper in a series of two which proves a version of a theorem of Harish-Chandra for quantum symmetric spaces in the maximally split case: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Here, we est...

in this paper we introduce the concept of generalized baer-invariant of a pair of groups with respect to two varieties ? and ? of groups. we give some inequalities for the generalized baer-invariant of a pair of finite groups, when ? is considered to be the schur-baer variety. further, we present a sufficient condition under which the order of the generalized baer-invariant of a pair of finite ...

We give a general procedure to construct algebro-geometric Feynman rules, that is, characters of the Connes–Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining motivic Feynman rules. We also con...

We partially generalize the known results on dp-minimal fields to dp-finite fields. prove a dichotomy: if K is sufficiently saturated expansion of field, then either has finite Morley rank or non-trivial Aut(K/A)-invariant valuation ring for some small set A. In positive characteristic case, we can even obtain henselian ring. Using this, classify pure

Generalized Cox’s construction associates with an algebraic variety a remarkable invariant – its total coordinate ring, or Cox ring. In this note we give a new proof of factoriality of the Cox ring when the divisor class group of the variety is finitely generated and free. The proof is based on a notion of graded factoriality. We show that if the divisor class group has torsion, then the Cox ri...

Let k be a field of finite characteristic p, and G a finite group acting on the left on a finite dimensional k-vector space V . Then the dual vector space V ∗ is naturally a right kG-module, and the symmetric algebra of the dual, R := Sym(V ∗), is a polynomial ring over k on which G acts naturally by graded algebra automorphisms, and if k is algebraically closed can be regarded as the space k[V...