Phùng Hô Hai

Tannaka-Krein duality for Hopf algebroids

We show that a Hopf algebroid can be reconstructed from a monoidal functor from a monoidal category into the category of rigid bimodules over a ring. We study the equivalence between the original category and the category of comodules over the reconstructed Hopf algebroid.

On Nori’s Fundamental Group Scheme

The aim of this note is to give a structure theorem on Nori’s fundamental group scheme of a proper connected variety defined over a perfect field and endowed with a rational point.

The Gauss-Manin connection and Tannaka duality

If f : X → Spec(K) is a smooth, geometrically connected variety defined over a field of characteristic 0, K ⊃ k is a field extension, and x ∈ X(K) is a rational point, one considers three Tannaka categories C(X/K), C(X/k), C(K/k) of flat connections with compatible fiber functors. The objects of C(X/K) are bundles (i.e., locally free coherent modules of finite type) with relative flat connections ((V,∇/K),∇/K : V → ΩX/K ⊗OX V), the ones of C(X/k) are bundles with flat absolute connections ((V,∇), ∇ : V → ΩX/k ⊗OX V), and the ones of C(K/k) are K-vector spaces with flat connections ((V,∇), ∇ : V → ΩK/k⊗KV). The morphisms are the flat morphisms and the fiber functor has values in the category of finite dimensional vector spaces VecK over K, defined by the restriction of V to x for C(X/K), C(X/k) and by V for C(K/k). Then C(X/K) is a neutral Tannaka category, and Tannaka duality [2,Theorem 2.11] yields the existence of a proalgebraic group schemeG(X/K) overK, so that C(X/K) becomes equivalent to the representation category Repf(G(X/K)) on finite dimensional K-vector spaces. The two other Tannaka categories C(X/k), C(K/k) are not necessarily neutral. We assume that they are defined over k, which is to say that k is the endomorphism ring EndC(K/k)((K, dK/k)) of the unit object,which in this case is the same as the subfield of K of flat sections. Equivalently, this is saying that k is algebraically closed in K. Then Tannaka duality [3, theoreme 1.12] yields the existence of groupoid schemes G(X/k), G(K/k) over k acting on Spec(K) ×k Spec(K), so that, in the groupoid sense, C(X/k) (resp., C(K/k)) becomes equivalent to the representation category Repf(K :

On the Poincaré series of quadratic algebras associated to Hecke symmetries

Hecke symmetries generalize the usual tensor symmetry of vector spaces

Tannakian duality over Dedekind rings and applications

v\otimes w\arrow w\otimes v

On an injectivity lemma in the proof of Tannakian duality

as well as the symmetry of vector superspaces. To a Hecke symmetry

The homological determinant of quantum groups of type A

R

Jacobi-Trudi Type Formula for Character of Irreducible Representations of \(\frak {gl}(m|1)\)

there associates a quadratic algebra which can be interpreted as the function algebra upon a certain quantum space. This paper investigates the Poincare series of this quadratic algebra. We showthat it is a rational function with numerator and denominator being a reciprocal polynomial and a skew-reciprocal polynomial, respectively.

On the flatness and the projectivity over Hopf subalgebras of Hopf algebras over Dedekind rings

We establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring. We use this duality and the known Tannakian duality due to Saavedra to study morphisms between flat affine group schemes. Next, we apply our new duality to the category of stratified sheaves on a smooth scheme over a Dedekind ring R to define the relative differential fundamental group scheme of the given scheme and compare the fibers of this group scheme with the fundamental group scheme of the fibers. When R is a complete DVR of equal characteristic we show that this category is Tannakian in the sense of Saavedra.

KOSZUL ALGEBRAS AND THE QUANTUM MACMAHON MASTER THEOREM

In this short work we give a very short and elementary proof of the injectivity lemma, which plays an important role in the Tannakian duality for Hopf algebras over a field. Based on this we provide some generalizations of this fact to the case of flat algebras over a noetherian domain.