A rigidity theorem for preLie algebras

Rigidity of quadratic Poisson tori

We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of ${\mathbb{N}}$-graded connected cluster algebras equipped with Gekhtman-Shapiro-Vainshtein Poisson structures. Based on this, we classify the groups of algebraic Poisson automorphisms of the open Schubert cells of the full flag varieties of semisimple algebraic groups over fields of characteristic 0, equipped with the standard Poisson structures. Their coor…

Flatness and Semi-Rigidity of Vertex Operator Algebras

In vertex operator algebra theories, most of the general theorems are proved under the assumptions of rationality and C_2-cofiniteness. In this paper, we obtain several general theorems without the assumption of rationality so that we can use them for proving rationality of given C_2-cofinite vertex operator algebras. For example, we apply them to orbifold models and show that if g is a finite automorphism of a rational C_2-cofinite vertex operator algebra T of CFT-type with T'\cong T and a fixed point subVOA T^g is C_2-cofinite, then T^g is also rational.

Flatness of Tensor Products and Semi-Rigidity for C_2-cofinite Vertex Operator Algebras I

We study properties of a C_2-cofinite vertex operator algebra of CFT type. If it is also rational and V'\cong V, then the rigidity of the tensor category of modules has been proved by Huang. When we treat an irrational C_2-cofinite VOA, the rigidity is too strong, because it is almost equivalent to be rational as we see. We introduce a natural weaker condition "semi-rigidity". Under this condition, we prove the following results. For a projective cover P of a V-module V and a finitely generated V-module M, the projective cover of M is a direct summand of the tensor product P\boxtimes M defined…

Braid Rigidity for Path Algebras

Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations of the braid groups $B_n$ for all $n\in \N$. We say that such representations are rigid if they are determined by the path algebra and the representations of $B_2$. We show that besides the known classical cases also the braid representations for the path algebra for the 7-dimensional representation of $G_2$ satisfies the rigidity condition, provided $B_3$ generates $\End(V^{\otimes 3})$. We obtain a complete classification …

Rigidity in higher representation theory

We describe a categorical g action, called a (g,theta) action, which is easier to check in practice. Most categorical g actions can be shown to be of this form. The main result is that a (g,theta) action carries actions of quiver Hecke algebras (KLR algebras). We discuss applications of this fact to categorical vertex operators, affine Grassmannians (or Nakajima quiver varieties) and to homological knot invariants.

一些简单仿射VOA的变形刚性

Deformation rigidity of some simple affine VOAs

本文证明了某些简单仿射VOA在特定条件下具有变形刚性，并推翻了$C_2$-余有限性和理性作为必要条件的假设。

Rigidity of K-theory under deformation quantization

Quantization, at least in some formulations, involves replacing some algebra of observables by a (more non-commutative) deformed algebra. In view of the fundamental role played by K-theory in non-commutative geometry and topology, it is of interest to ask to what extent K-theory remains "rigid" under this process. We show that some positive results can be obtained using ideas of Gabber, Gillet-Thomason, and Suslin. From this we derive that the algebraic K-theory with finite coefficients of a deformation quantization of the functions on a compact symplectic manifold, forgetting the topology, re…

Grothendieck-Verdier范畴中的交换代数、刚性与顶点算子代数

Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras

研究Grothendieck-Verdier范畴中交换代数的模范畴及其局部子范畴的刚性继承问题，给出刚性判定条件，应用于顶点算子代数扩张的强有理性证明，降低相关证明难度。