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Invariants of finite group schemes
 J. London Math. Soc. 65, Part
"... Let k be an algebraically closed field, G a finite group scheme over k operating on a scheme X over k. Under assumption that X can be covered by Ginvariant affine open subsets the classical results in [3] and [14] describe the quotient X/G. In case of a free action X is known to be a principal homo ..."
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Let k be an algebraically closed field, G a finite group scheme over k operating on a scheme X over k. Under assumption that X can be covered by Ginvariant affine open subsets the classical results in [3] and [14] describe the quotient X/G. In case of a free action X is known to be a principal homogeneous Gspace over X/G.
Lagrange’s theorem for Hopf monoids in species
, 2012
"... Abstract. Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras, ..."
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Abstract. Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras,
THE GAUSSMANIN CONNECTION AND TANNAKA DUALITY
, 2005
"... Abstract. If f: X → S is a fibration of complex connected analytic manifolds, given a point x ∈ X, there is an exact sequence of fundamental groups 0 → π1(Xs, x) → π1(X, x) → π1(S, f(x)) → 0. Similarly, if f: X → Spec(Fq) is a smooth, geometrically connected variety defined over a finite field, gi ..."
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Abstract. If f: X → S is a fibration of complex connected analytic manifolds, given a point x ∈ X, there is an exact sequence of fundamental groups 0 → π1(Xs, x) → π1(X, x) → π1(S, f(x)) → 0. Similarly, if f: X → Spec(Fq) is a smooth, geometrically connected variety defined over a finite field, given a point x ∈ X ( ¯ Fq), there is an exact sequence of étale fundamental groups 0 → π1(X ×Fq ¯ Fq, x) → π1(X, x) → Gal ( ¯ Fq/Fq) → 0. In particular, for any representation ρ (resp. ℓadic representation) of π1(X, x), one obtains an action of π1(S, f(x)), resp. Gal ( ¯ Fq/Fq), on Hi (π1(Xs, x), ρ), resp. Hi (π1(X ¯Fq, ×Fq x), ρ). When the natural map Hi (π1(Xs, x), ρ) → Hi (Xs, ρ), resp. Hi ¯Fq, (π1(X×Fq x), ρ) → Hi (X ×Fq ¯ Fq, ρ) is an isomorphism, this action defines the cohomology of the fibers as a local system over the base, resp. a Galois action on the cohomology of ρ over X ¯Fq. ×Fq A good analog in algebraic geometry of the topological fundamental group on one side and the étale fundamental group on the other side is provided by the Tannaka group associated to the category of flat connections. The action of the fundamental group or of the Galois group of the base corresponds to the GaußManin connection. The purpose of this article is to show that the analogy isn’t straightforward, that those actions are difficult to define, partly because the homomorphism analogous to π1(Xs, x) → π1(X, x) is not injective. 1.
Derivations with quantum group action
"... Abstract. The derivations of a left coideal subalgebra B of a Hopf algebra A which are compatible with the comultiplication of A (that is, the covariant first order differential calculi, as defined by Woronowicz, on a quantum homogeneous space) are related to certain right ideals of B. The correspon ..."
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Abstract. The derivations of a left coideal subalgebra B of a Hopf algebra A which are compatible with the comultiplication of A (that is, the covariant first order differential calculi, as defined by Woronowicz, on a quantum homogeneous space) are related to certain right ideals of B. The correspondence is onetoone if A is faithfully flat as a right Bmodule. This generalizes the result for B = A due to Woronowicz. A definition for the dimension of a first order differential calculus at a classical point is given. For the quantum 2sphere S 2 qc of Podle´s under the assumptions qn+1 ̸ = 1 and c ̸ = −q2n /(q2n + 1) 2 for all n = 0, 1,..., three 2dimensional covariant first order differential calculi exist if c = 0, one exists if c = ∓q/(±q + 1) 2 and none else. This extends a result of Podle´s. 1. PRELIMINARIES A derivation of an algebra B over C (the complex numbers) is defined
Relative invariants, difference equations, and the PicardVessiot theory (revised version 5)
, 2006
"... to learn mathematics from the beginning when I just started afresh my life. About four years ago he suggested to study on archimedean local zeta functions of several variables as my first research task for the Master’s thesis. I can not give enough thanks to his heartwarming encouragement all over ..."
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to learn mathematics from the beginning when I just started afresh my life. About four years ago he suggested to study on archimedean local zeta functions of several variables as my first research task for the Master’s thesis. I can not give enough thanks to his heartwarming encouragement all over the period of this program. I thank Professor T. Kogiso and Professor K. Sugiyama for being close advisers on prehomogeneous vector spaces. I also would like to thank Professor M. Takeuchi and Professor A. Masuoka. In June or July 2003, I had a chance to know the existence of Professor Takeuchi’s paper [2] when I was trying to understand [1]. Since I once attended to an introductory part of his lecture on Hopf algebras at 2002, the paper seemed like just what I wanted. Then I was absorbed in it and came to think that the results can be extended to involve [1]. But I could not have developed it in the presented form without the collaboration with Professor Masuoka. He suggested the viewpoint from relative Hopf modules as is described in Section 3.2, added many interesting results, and helped me to complete several proofs. Finally I would like to thank my parents for supporting my decision to study mathematics as the lifework.
GENERALIZED HOPF MODULES FOR BIMONADS
"... Abstract. Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler’s Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodulemonad ..."
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Abstract. Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler’s Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodulemonad and a algebracomonoid over it. As an application we obtain a generalization of another classical theorem from the Hopf algebra literature, due to Schneider, which itself is an extension of Sweedler’s result (to the setting of Hopf Galois extensions).
Acknowledgements i
, 2005
"... First of all I would like to thank my supervisor Professor T. Kimura. He taught me how to learn mathematics from the beginning when I just started afresh my life. About four years ago he suggested to study on archimedean local zeta functions of several variables as my first research task for the Mas ..."
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First of all I would like to thank my supervisor Professor T. Kimura. He taught me how to learn mathematics from the beginning when I just started afresh my life. About four years ago he suggested to study on archimedean local zeta functions of several variables as my first research task for the Master’s thesis. I can not give enough thanks to his heartwarming encouragement all over the period of this program. I thank Professor T. Kogiso and Professor K. Sugiyama for being close advisers on prehomogeneous vector spaces. I also would like to thank Professor M. Takeuchi and Professor A. Masuoka. In June or July 2003, I had a chance to know the existence of the Professor Takeuchi’s paper [2] when I was trying to understand [1]. Since I once attended to an introductory part of his lecture on Hopf algebras at 2002, the paper seemed like just what I wanted. Then I was absorbed in it and came to think that the results can be extended to involve [1]. But I could not have developed it in the presented form without the collaboration with Professor Masuoka. He suggested the viewpoint from relative Hopf modules as is described in Section 3.2, added many interesting results, and helped me to complete several proofs. Finally I would like to thank my parents for supporting my decision to study mathematics as the lifework. References [1] M. van der Put, M.F. Singer, “Galois theory of difference equations”, Lecture Notes in Math. 1666,
A CONSTRUCTION OF A QUOTIENT TENSOR CATEGORY PHÙNG HÔ HAI
, 2008
"... Abstract. For a rigid tensor abelian category T over a field k we introduce a notion of a normal quotient q: T − → Q. In case T is a Tannaka category, our notion is equivalent to Milne’s notion of a normal quotient [5]. More precisely, if T is the category of finite dimensional representations of a ..."
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Abstract. For a rigid tensor abelian category T over a field k we introduce a notion of a normal quotient q: T − → Q. In case T is a Tannaka category, our notion is equivalent to Milne’s notion of a normal quotient [5]. More precisely, if T is the category of finite dimensional representations of a groupoid scheme G over k, then Q is equivalent to the representation category of a normal subgroupoid scheme of G. We describe such a quotient in terms of the subcategory S of T consisting of objects which become trivial in Q. We show that, under some condition on S, Q is uniquely determined by S. If S is a finite tensor category, we show that the quotient of T by S exists. In particular we show the existence of the base change of T with respect to finite field extensions. As an application, we obtain a condition for the exactness of sequences of groupoid schemes in terms of the representation categories.
ON A DISCREPANCY AMONG PICARDVESSIOT THEORIES IN POSITIVE CHARACTERISTICS
, 2006
"... Abstract. There is a serious discrepancy among literature on the PicardVessiot theory in positive characteristics (for iterative differential fields). We consider descriptions of Galois correspondence in four approaches to this subject: Okugawa’s result [7], Takeuchi’s Hopf algebraic approach [11] ..."
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Abstract. There is a serious discrepancy among literature on the PicardVessiot theory in positive characteristics (for iterative differential fields). We consider descriptions of Galois correspondence in four approaches to this subject: Okugawa’s result [7], Takeuchi’s Hopf algebraic approach [11] (and [3]), the result of Matzat and van der Put [6], and the model theoretic approach by Pillay [8]. In the three approaches except Takeuchi’s one, Galois correspondence is described between closed subgroups of algebraic matrix groups and its fixed fields. But such a description has a problem that the Galois correspondence may not be bijective there. We explain this problem in the first section by giving an explicit example. We should use affine group schemes instead of algebraic matrix groups to obtain a suitable Galois correspondence, as in Takeuchi’s approach. But intermediate fields are not necessarily fixed fields there. In the second section, we give some sufficient conditions for intermediate artinian simple module algebras to be fixed algebras in the context of the unified PicardVessiot theory developed by the author and Masuoka [3].
A CONSTRUCTION OF A QUOTIENT TENSOR CATEGORY
, 2006
"... Abstract. Let f: G → A be a surjective homomorphism of transitive groupoid schemes and let L denote the kernel of f. The exact sequence of groupoid schemes 1 → L → G → A → 1 induces a sequence of functors between the categories of finite representations of these groupoid schemes ..."
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Abstract. Let f: G → A be a surjective homomorphism of transitive groupoid schemes and let L denote the kernel of f. The exact sequence of groupoid schemes 1 → L → G → A → 1 induces a sequence of functors between the categories of finite representations of these groupoid schemes