Automorphic vector bundles on the stack of G-zips

For a connected reductive group G over a finite field, we study automorphic vector bundles on the stack of G-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski-Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of G-zips and a category of admissible modules with actions of a 0-dimensional algebraic subgroup a Levi subgroup and monodromy operators.

Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In the first part of the second section we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin’s problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. The second part of the second section is dedicated to particular cases of Plotkin’s problem. The last part of the second section is devoted to Plotkin’s problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last sections deal with algorithmic problems for noncommutative and commutative algebraic geometry.The first part of it is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. The second part of the last section is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given.

Shtukas with one leg III

This chapter presents a third lecture on one-legged shtukas. The goal is to complete the proof of Fargues' theorem. To complete the proof of Fargues' theorem, it remains to prove the following result, where Y = Spa Ainf REVERSE SOLIDUS {xk}. Theorem 14.2.1 posits that there is an equivalence of categories between finite free Ainf-modules and vector bundles on Y. One should think of this as being an analogue of a classical result: If (R, m) is a 2-dimensional regular local ring, then finite free R-modules are equivalent to vector bundles on (Spec R)REVERSE SOLIDUS {m}. The chapter then provides a proof of Theorem 14.2.1.

Graded Algebraic Theories

We provide graded extensions of algebraic theories and Lawvere theories that correspond to graded monads. We prove that graded algebraic theories, graded Lawvere theories, and finitary graded monads are equivalent via equivalence of categories, which extends the equivalence for monads. We also give sums and tensor products of graded algebraic theories to combine computational effects as an example of importing techniques based on algebraic theories to graded monads.

Lie theory for symmetric Leibniz algebras

Lie algebras and groups equipped with a multiplication $$\mu $$μ satisfying some compatibility properties are studied. These structures are called symmetric Lie $$\mu $$μ-algebras and symmetric $$\mu $$μ-groups respectively. An equivalence of categories between symmetric Lie $$\mu $$μ-algebras and symmetric Leibniz algebras is established when 2 is invertible in the base ring. The second main result of the paper is an equivalence of categories between simply connected symmetric Lie $$\mu $$μ-groups and finite dimensional symmetric Leibniz algebras.

The Category of Ordered Bratteli Diagrams

A category structure for ordered Bratteli diagrams is proposed in which isomorphism coincides with the notion of equivalence of Herman, Putnam, and Skau. It is shown that the natural one-to-one correspondence between the category of Cantor minimal systems and the category of simple properly ordered Bratteli diagrams is in fact an equivalence of categories. This gives a Bratteli–Vershik model for factor maps between Cantor minimal systems. We give a construction of factor maps between Cantor minimal systems in terms of suitable maps (called premorphisms) between the corresponding ordered Bratteli diagrams, and we show that every factor map between two Cantor minimal systems is obtained in this way. Moreover, solving a natural question, we are able to characterize Glasner and Weiss’s notion of weak orbit equivalence of Cantor minimal systems in terms of the corresponding C*-algebra crossed products.

Equivariant perverse sheaves on Coxeter arrangements and buildings

When $W$ is a finite Coxeter group acting by its reflection representation on $E$, we describe the category ${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ of $W$-equivariant perverse sheaves on $E_{\mathbb C}$, smooth with respect to the stratification by reflection hyperplanes. By using Kapranov and Schechtman's recent analysis of perverse sheaves on hyperplane arrangements, we find an equivalence of categories from ${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ to a category of finite-dimensional modules over an algebra given by explicit generators and relations. We also define categories of equivariant perverse sheaves on affine buildings, e.g., $G$-equivariant perverse sheaves on the Bruhat--Tits building of a $p$-adic group $G$. In this setting, we find that a construction of Schneider and Stuhler gives equivariant perverse sheaves associated to depth zero representations. Comment: 28 pages, 6 figures. v5 processed for publication in Epiga

Complex manifolds with maximal torus actions

In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples {(\Delta,\mathfrak{h},G)} of nonsingular complete fan Δ in {\mathfrak{g}}, complex vector subspace {\mathfrak{h}} of {\mathfrak{g}^{\mathbb{C}}} and compact torus G satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate {\mathbb{C}^{n}}-actions on compact connected complex manifolds of complex dimension n, and construction of concrete examples of non-Kähler manifolds.

Virtual tangles and fiber functors

We define a category [Formula: see text] of tangles diagrams drawn on surfaces with boundaries. On the one hand, we show that there is a natural functor from the category of virtual tangles to [Formula: see text] which induces an equivalence of categories. On the other hand, we show that [Formula: see text] is universal among ribbon categories equipped with a strong monoidal functor to a symmetric monoidal category. This is a generalization of the Shum–Reshetikhin–Turaev theorem characterizing the category of ordinary tangles as the free ribbon category. This gives a straightforward proof that all quantum invariants of links extend to framed oriented virtual links. This also provides a clear explanation of the relation between virtual tangles and Etingof–Kazhdan formalism suggested by Bar-Natan. We prove a similar statement for virtual braids, and discuss the relation between our category and knotted trivalent graphs.