We present a simple unified formula expressing the denominators of normalized R-matrices between fundamental modules over quantum loop algebras type ADE. It has an interpretation in terms representations Dynkin quivers and can be proved way using geometry graded quiver varieties. As by-product, we obtain geometric Kang-Kashiwara-Kim's generalized affine Schur-Weyl duality functor when it arises...

Let ? be a commutative ring with involution ? containing an element ? such that ?+??=1 and let Alg?? the category of ?-algebras equipped semilinear preserving homomorphisms. We construct triangulated kkh functor jh:Alg???kkh is homotopy invariant, matricially hermitian stable excisive universal initial these properties. prove version Karoubi's fundamental theorem holds in kkh. By property latte...

Let G be a finite group. It is well known that a Mackey functor {H 7→ M(H)} is a module over the Burnside ring functor {H 7→ Ω(H)}, where H ranges over the set of all subgroups of G. For a fixed homomorphism w : G → {−1, 1}, the Wall group functor {H 7→ Ln(Z[H], w|H)} is not a Mackey functor if w is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside rin...

The mirror functor as constructed by Polishchuk and Zaslow depends on various choices. This calls for a description of how the constructed functor depends on these choices. For a “good” mirror functor Φτ we would expect that we get a symplectic counterpart, that has a geometric explanation in terms of complexified symplectic tori. But unfortunately, the mirror functor does not have this expecte...

As a fundamental notion, the free differential algebra on set is concretely constructed as polynomial variables. Such construction not known for more general notion of an algebra, from left adjoint functor forgetful algebras to algebras, instead sets. In this paper we show that generator-relation properties base can be extended algebra. More precisely, Gr\"obner-Shirshov basis property allowing...

1 Let G be a finite group. The Burnside ring B(G) of the group G is one of the fundamental representation rings of G, namely the ring of permutation representations. It is in many ways the universal object to consider when looking at the category of G-sets. It can be viewed as an analogue of the ring Z of integers for this category. It can be studied from different points of view. First B(G) is...