Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M/G. Guillemin and Sternberg [Invent. Math. 67 (1982), 515–538] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M/G and the G-invariant subspa...

Let A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x), B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous f...

Let K be a general nite commutative ring. We refer to a family gn, n = 1, 2, . . . of bijective polynomial multivariate maps of K as a family with invertible decomposition gn = g ng 2 n . . . g k n, such that the knowledge of the composition of g n allows computation of g i n for O(n ) (s > 0) elementary steps. A polynomial map g is stable if all non-identical elements of kind g, t > 0 are of t...

We consider a family of finitely presented groups, called Universal Left Invertible Element (or ULIE) groups, that are universal for existence of one–sided invertible elements in a group ring K[G], where K is a field or a division ring. We show that for testing Kaplansky’s Direct Finiteness Conjecture, it suffices to test it on ULIE groups, and we show that there is an infinite family of non-am...

In the last lecture we have learned that the category of modules over a braided Hopf algebra H is a braided monoidal category. A braided Hopf algebra is a rather sophisticated algebraic object, it is not easy to give interesting nontrivial examples. In this text we develop a theory that will lead to a concrete recipe which produces a nontrivial braided Hopf algebra D(A) (called Drinfeld’s quant...

In this paper, we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois their character tables. Based on these symmetries, generalize well-known result Burnside from representation theory finite groups. More precisely, any row corresponding to non-invertible object in table weakly category contains zero entry.

We classify various types of graded extensions a finite braided tensor category $$\mathcal {B}$$ in terms its 2-categorical Picard groups. In particular, we prove that by group A correspond to monoidal 2-functors from the (consisting invertible central -module categories). Such functors can be expressed Eilnberg-Mac Lane cohomology. describe detail groups symmetric fusion categories and pointed...

On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension at least three. We then prove that if non-empty the space of metrics with invertible Dirac operators is disconnected in dimensions n ≡ 0, 1, 3, 7 mod 8, n ≥...

In this paper, we give a proof of homological mirror symmetry for two variable invertible polynomials, where the group on $B$-side is taken to be maximal. The involves an explicit gluing construction Milnor fibres, and as application, prove derived equivalences between certain nodal stacky curves, some whose irreducible components have non-trivial generic stabiliser.

In this work we study the skew group algebra Λ[G] when G is a finite group acting on Λ whose order is invertible in Λ. Here, we assume that Λ is a finite-dimensional algebra over an algebraically closed field k. The aim is to describe all possible actions of a finite abelian group on an hereditary algebra of finite or tame representation type, to give a description of the resulting skew group a...