39,297 research outputs found
Connes-Moscovici characteristic map is a Lie algebra morphism
Let be a Hopf algebra with a modular pair in involution (\Character,1).
Let be a (module) algebra over equipped with a non-degenerated
\Character-invariant -trace . 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
of Batalin-Vilkovisky algebras from the cotorsion product of ,
, to the Hochschild cohomology of ,
. Let 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
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
Lusztig has constructed a Frobenius morphism for quantum groups at an
-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
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 be a Hopf algebra with a modular pair in involution (\Character,1). Let be a (module) algebra over equipped with a non-degenerated \Character-invariant -trace . 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 of Batalin-Vilkovisky algebras from the cotorsion product of , , to the Hochschild cohomology of , . Let 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 is injective
The Period-Index Problem of the Canonical Gerbe of Symplectic and Orthogonal Bundles
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 -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
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
Let be an integral -algebra of finite type over an algebraically
closed field of characteristic . Given a collection of
-derivations on , that we interpret as algebraic vector fields on
, we study the group spanned by the hypersurfaces of
invariant for modulo the rational first integrals of .
We prove that this group is always a finite -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 -algebra between and
, we show that the kernel of the pull-back morphism is a finite -vector space. In particular, if is a
UFD, then the Picard group of is finite.Comment: 16 page
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