Transformation Groups最新文献

For a smooth projective curve X and reductive group G, the Whittaker functional on nilpotent sheaves on (Bun _G(X)) is expected to correspond to global sections of coherent sheaves on the spectral side of Betti geometric Langlands. We prove that the Whittaker functional calculates the shifted microstalk of nilpotent sheaves at the point in the Hitchin moduli where the Kostant section intersects the global nilpotent cone. In particular, the shifted Whittaker functional is exact for the perverse t-structure and commutes with Verdier duality. Our proof is topological and depends on the intrinsic local hyperbolic symmetry of (Bun _G(X)). It is an application of a general result relating vanishing cycles to the composition of restriction to an attracting locus followed by vanishing cycles.

We study a family of tessellations of the Euclidean plane which are obtained by local developments of algebraic equations satisfied by the Pauli matrices.

Let K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor₀(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor₀(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.

Let H be a Hopf algebra that is $\mathbb{Z}$ -graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of H to be a Zhang twist of H. In particular, we introduce the notion of a twisting pair for H such that the Zhang twist of H by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated with quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction.

Flag domains are open orbits of noncompact real forms of complex semisimple Lie groups acting on flag manifolds. To each flag domain one can associate a compact complex manifold called the base cycle. The ampleness of the normal bundle of the base cycle in a flag domain measures the concavity near the base cycle. In this paper we compute the ampleness of normal bundles of base cycles in flag domains in various cases, including flag domains in the full flag manifolds G/B when G is classical, and period domains parameterizing polarized Hodge structures with fixed Hodge numbers.

Quasifolds are spaces that are locally modelled by quotients of (mathbb {R}^n) by countable affine group actions. These spaces first appeared in Elisa Prato’s generalization of the Delzant construction, and special cases include leaf spaces of irrational linear flows on the torus, and orbifolds. We consider the category of diffeological quasifolds, which embeds in the category of diffeological spaces, and the bicategory of quasifold groupoids, which embeds in the bicategory of Lie groupoids, (right-)principal bibundles, and bibundle morphisms. We prove that, restricting to those morphisms that are locally invertible, and to quasifold groupoids that are effective, the functor taking a quasifold groupoid to its diffeological orbit space is an equivalence of the underlying categories. These results complete and extend earlier work with Masrour Zoghi.

We construct a non-trivial homomorphism from the Guay’s affine Yangian associated with (widehat{mathfrak {sl}}(n)) to the universal enveloping algebra of the W-algebra associated with a Lie algebra (mathfrak {gl}(m+n)) and a nilpotent element of type ((2^{n},1^{m-n})) for (m>n).

Let (X={text {Spec}};B) be a factorial affine variety defined over an algebraically closed field k of characteristic zero with a nontrivial action of the additive group (G_a) associated to a locally nilpotent derivation (delta ) on B. In this article, we study X of dimension (ge 3) under the assumption that the plinth ideal (text {pl}(delta )=delta (B)cap A) is contained in an ideal (alpha A) generated by a prime element (alpha in A={text {Ker}},delta ). Suppose that (A={text {Ker}},delta ) is an affine k-domain. The quotient morphism (pi : X rightarrow Y={text {Spec}};A) splits to a composite (textrm{pr} circ p) of the projection (textrm{pr}: Ytimes mathbb A^1 rightarrow Y) and a (G_a)-equivariant birational morphism (p: X rightarrow Ytimes mathbb A^1) where (G_a) acts on (mathbb A^1) by translation. By decomposing (p: X rightarrow Ytimes mathbb A^1) to a sequence of (G_a)-equivariant affine modifications, we investigate the structure of X. We also show that the general closed fiber of (pi ) over the closed set (V(alpha )={text {Spec}};A/alpha A) consists of a disjoint union of m affine lines where (mge 2).

For an arbitrary fivefold ramified covering (varvec{f :Xrightarrow Y}) between compact Riemann surfaces, each possible Galois closure (varvec{hat{f}:hat{X}rightarrow Y}) is determined in terms of the branching data of (varvec{f}). Since (varvec{{{,textrm{Mon},}}(f)}) acts on (varvec{hat{f}}), it also acts on the Jacobian variety (varvec{{{,textrm{J},}}(X)}), and we describe its group algebra decomposition in terms of the Jacobian and Prym varieties of the intermediate coverings of (varvec{hat{f}}). The dimension and induced polarization of each abelian variety in the decomposition is computed in terms of the branching data of (varvec{f}).

In this paper, we obtain a complete classification of compact hyperbolic Coxeter five-dimensional polytopes with nine facets.