39,297 research outputs found

    Connes-Moscovici characteristic map is a Lie algebra morphism

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    Let HH be a Hopf algebra with a modular pair in involution (\Character,1). Let AA be a (module) algebra over HH equipped with a non-degenerated \Character-invariant 11-trace τ\tau. We show that Connes-Moscovici characteristic map \varphi_\tau:HC^*_{(\Character,1)}(H)\rightarrow HC^*_\lambda(A) is a morphism of graded Lie algebras. We also have a morphism Φ\Phi of Batalin-Vilkovisky algebras from the cotorsion product of HH, CotorH∗(k,k)\text{Cotor}_H^*({\Bbbk},{\Bbbk}), to the Hochschild cohomology of AA, HH∗(A,A)HH^*(A,A). Let KK be both a Hopf algebra and a symmetric Frobenius algebra. Suppose that the square of its antipode is an inner automorphism by a group-like element. Then this morphism of Batalin-Vilkovisky algebras Φ:CotorK∨∗(F,F)≅ExtK(F,F)↪HH∗(K,K)\Phi:\text{Cotor}_{K^\vee}^*(\mathbb{F},\mathbb{F})\cong \text{Ext}_{K}(\mathbb{F},\mathbb{F}) \hookrightarrow HH^*(K,K) is injective.Comment: submitted version. Corollary 28 and Section 9 has been added. Section 9 computes the Batalin-Vilkovisky algebra on the rational cotor of an universal envelopping algebra of a lie algebr

    Hall algebras and the Quantum Frobenius

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    Lusztig has constructed a Frobenius morphism for quantum groups at an â„“\ell-th root of unity, which gives an integral lift of the Frobenius map on universal enveloping algebras in positive characteristic. Using the Hall algebra we give a simple construction of this map for the positive part of the quantum group attached to an arbitrary Cartan datum in the nondivisible case.Comment: Final version: typos corrected and exposition improve

    Connes-Moscovici characteristic map is a Lie algebra morphism

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    submitted version. Corollary 28 and Section 9 has been added. Section 9 computes the Batalin-Vilkovisky algebra on the rational cotor of an universal envelopping algebra of a lie algebra.Let HH be a Hopf algebra with a modular pair in involution (\Character,1). Let AA be a (module) algebra over HH equipped with a non-degenerated \Character-invariant 11-trace τ\tau. We show that Connes-Moscovici characteristic map \varphi_\tau:HC^*_{(\Character,1)}(H)\rightarrow HC^*_\lambda(A) is a morphism of graded Lie algebras. We also have a morphism Φ\Phi of Batalin-Vilkovisky algebras from the cotorsion product of HH, CotorH∗(k,k)\text{Cotor}_H^*({\Bbbk},{\Bbbk}), to the Hochschild cohomology of AA, HH∗(A,A)HH^*(A,A). Let KK be both a Hopf algebra and a symmetric Frobenius algebra. Suppose that the square of its antipode is an inner automorphism by a group-like element. Then this morphism of Batalin-Vilkovisky algebras Φ:CotorK∨∗(F,F)≅ExtK(F,F)↪HH∗(K,K)\Phi:\text{Cotor}_{K^\vee}^*(\mathbb{F},\mathbb{F})\cong \text{Ext}_{K}(\mathbb{F},\mathbb{F}) \hookrightarrow HH^*(K,K) is injective

    The Period-Index Problem of the Canonical Gerbe of Symplectic and Orthogonal Bundles

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    We consider regularly stable parabolic symplectic and orthogonal bundles over an irreducible smooth projective curve over an algebraically closed field of characteristic zero. The morphism from the moduli stack of such bundles to its coarse moduli space is a μ2\mu_2-gerbe. We study the period and index of this gerbe, and solve the corresponding period-index problem.Comment: 19 pages. Complete rewrite of the previous version, including expanded results on the moduli of parabolic G-bundles. To appear in the Journal of Algebra. Comments welcom

    Relative Calabi-Yau structures and ice quivers with potential

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    In 2015, Van den Bergh showed that complete 3-Calabi-Yau algebras over an algebraically closed field of characteristic 0 are equivalent to Ginzburg dg algebras associated with quivers with potential. He also proved the natural generalisation to higher dimensions and non-algebraically closed ground fields. The relative version of the notion of Ginzburg dg algebra is that of Ginzburg morphism. For example, every ice quiver with potential gives rise to a Ginzburg morphism. We generalise Van den Bergh's theorem by showing that, under suitable assumptions, any morphism with a relative Calabi-Yau structure is equivalent to a Ginzburg(-Lazaroiu) morphism. In particular, in dimension 3 and over an algebraically closed ground field of characteristic 0, it is given by an ice quiver with potential. Thanks to the work of Bozec-Calaque-Scherotzke, this result can also be viewed as a non-commutative analogue of Joyce-Safronov's Lagrangian neighbourhood theorem in derived symplectic geometry.Comment: 39 pages; v2: more accurate historical account in introduction, reference to Joyce-Safronov's work added, many minor change

    Invariant hypersurfaces for derivations in positive characteristic

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    Let AA be an integral kk-algebra of finite type over an algebraically closed field kk of characteristic p>0p>0. Given a collection D{\cal{D}} of kk-derivations on AA, that we interpret as algebraic vector fields on X=Spec(A)X=Spec(A), we study the group spanned by the hypersurfaces V(f)V(f) of XX invariant for D{\cal{D}} modulo the rational first integrals of D{\cal{D}}. We prove that this group is always a finite Z/p\mathbb{Z}/p-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant for a foliation of codimension 1. As an application, given a kk-algebra BB between ApA^p and AA, we show that the kernel of the pull-back morphism Pic(B)→Pic(A)Pic(B)\rightarrow Pic(A) is a finite Z/p\mathbb{Z}/p-vector space. In particular, if AA is a UFD, then the Picard group of BB is finite.Comment: 16 page
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