代数表示论青年论坛

I will present joint work with Quoc P. Ho on our theory of graded sheaves, which offers a uniform construction of "mixed versions" or "graded lifts" in the sense of Beilinson-Ginzburg-Soergel applicable to arbitrary Artin stacks. As a straightforward reinterpretation of classical results, this framework allows for the categorification of many quantum algebras, including Hecke algebras and positive quantum groups, through graded sheaves. In a new application, we introduce a graded version of Lusztig's character sheaves and utilize it to prove a conjecture by Gorsky, Negut, and Rasmussen, which establishes a connection between the HOMFLY homology of knots and links and the cohomology of coherent sheaves on Hilbert schemes.

Semi-orthogonal decomposition serves as a powerful tool for understanding derived categories. Through the additivity theorem, it also aids in computing (higher) algebraic K-theory. Additionally, Hermitian K-theory—a quadratic refinement of algebraic K-theory—can be approached using semi-orthogonal decomposition, with the main focus on understanding the symmetry of semi-orthogonal decompositions under various dualities in derived categories. In this talk, I will explain how semi-orthogonal decomposition can be utilized to understand the connecting homomorphism in the localization sequence of Hermitian K-theory and present several computations employing this approach.

It is proved that any almost tilting module over a gentle algebra is partial tilting, that is, it can be completed as a tilting module. Furthermore, it has at most $2n$ complements, which confirms a （deformed）conjecture of Happel for the case of gentle algebras. At the same time, for any $n > 2$ and $0< m < n-1$, there always exists a connected gentle algebra with rank n and a pre-tilting module over it with rank m which is not partial tilting. The tool we use is the surface model associated with the module category of a gentle algebra. The main technique is doing inductions by cutting the surface, which is expected to be useful elsewhere.

The aim of my talk (based on joint work in progress with Arkady Berenstein and Vladimir Retakh) is to introduce and study certain noncommutative algebras A for any marked orbifold. These algebras admit noncommutative clusters, i.e., embeddings of a given group G which is either free or one-relator (we call it triangle group) into the multiplicative monoid A×. The clusters are parametrized by triangulations of the orbifold and exhibit a noncommutative Laurent Phenomenon, which asserts that generators of the algebra can be written as sums of the images of elements of G for any noncommutative cluster. If the surface is unpunctured, then our algebra A can be specialized to the ordinary quantum cluster algebra, and the noncommutative Laurent Phenomenon becomes the (positive) quantum one. It turns out that there is a natural action of a certain braid-like group Br_A by automorphisms of G on each cluster in a compatible way (this is, indeed, the braid group Br_n if the surface is an unpunctured disk with n+2 marked boundary points). If surface is punctured, the algebra A admits a family of commuting automorphisms which will give new clusters and new "tagged" noncommutative Laurent Phenomena. There are important elements in A assigned to each marked point, which we refer to as noncommutative angles (or h-lengths). They belong to the group algebra of each cluster group and are invariant under all noncommutative cluster mutations. This eventually gives rise to noncommutative integrable systems on unpunctured cylinders and other surfaces which, in particular, recover the ones introduced by Kontsevich in 2011 together with their Laurentness and positivity.

In this talk, I present our recent results on affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(l,0)$ at admissible level $l$. We prove that the category of weak $L_{\widehat{osp(1|2)}}(l,0)$-modules on which the positive part of $\widehat{osp(1|2)}$ acts locally nilpotently is semisimple. Then we prove that Q-graded vertex operator superalgebras $(L_{\widehat{osp(1|2)}}(l,0),\omega_\xi)$ with a new Virasoro element $\omega_\xi$ are rational and the irreducible modules are exactly the admissible modules for $\widehat{osp(1|2)}$, where $0<\xi<1$ is a rational number. Furthermore, we determine the Zhu's algebras $A(L_{\widehat{osp(1|2)}}(l,0))$ and their bimodules $A(L(l,j))$ for $(L_{\widehat{osp(1|2)}}(l,0),\omega_\xi)$, where $j$ is the admissible weight. As an application, we calculate the fusion rules among the irreducible ordinary modules of $(L_{\widehat{osp(1|2)}}(l,0),\omega_\xi)$. This is a joint work with Huaimin Li.

Guided by Koszul duality theory, we consider the graded Lie algebra of coderivations of the cofree conilpotent graded cocommutative cotrialgebra generated by a graded vector space V. We show that in the case of V being a shift of an ungraded vector space W, Maurer-Cartan elements of this graded Lie algebra are exactly post-Lie algebra structures on W. The cohomology of a post-Lie algebra is then defined using Maurer-Cartan twisting. The second cohomology group of a post-Lie algebra has a familiar interpretation as equivalence classes of infinitesimal deformations. Next we define a post-Lie-infty algebra structure on a graded vector space to be a Maurer-Cartan element of the aforementioned graded Lie algebra. Post-Lie-infty algebras admit a useful characterization in terms of L-infty-actions (or open-closed homotopy Lie algebras). Finally, we introduce the notion of homotopy Rota-Baxter operators on open-closed homotopy Lie algebras and show that certain homotopy Rota-Baxter operators induce post-Lie-infty algebras. This is a joint work with Andrey Lazarev and Rong Tang.

Coideal subalgebras of Hopf algebras are conceptually the right context to define and understand quantum homogeneous spaces. The class of pointed Hopf algebras with abelian coradical includes many important examples of Hopf algebras. In this talk, I will present several structure results for coideal subalgebras of pointed Hopf algebras with abelian coradical under some mild conditions.

In 2018, in their study of deformations of spaces of stability conditions, Ikeda-Qiu introduced $\mathbb{X}$-Calabi-Yau completions (=bigraded CY-completions) and proved that the cluster category associated with the $\mathbb{X}$-CY-completion of a Dynkin quiver is equivalent to the bounded derived category of its category of representations. We will report on joint work with Fan Li and Yu Qiu where we generalize this theorem to suitable differential graded algebras. We will then compare this result with Hanihara's recent generalization to dg algebras of Happel's description of the derived category as the singularity category of the repetitive algebra.

Approximable triangulated categories, as introduced and developed by Neeman, provide a solid framework for studying localization sequences within triangulated categories. In this talk, we demonstrate that a recollement of approximable triangulated categories induces interesting short exact sequences of both triangulated subcategories and Verdier quotient categories. Additionally, we discuss applications of these results within the derived categories of finite-dimensional algebras, DG algebras, and schemes. This is based on a joint work with Yongliang Sun.

In this talk, we introduce the notion of $d$-hearts, which is a finite analogue of hearts of bounded $t$-structures, and it can be regarded as a counterpart of $d$-cluster tilting subcategories in negative Calabi-Yau triangulated categories and as a generalization of simple-minded systems. As an analogue of classical result due to Beilinson-Bernstein-Deligne, we show that $d$-hearts are abelian categories for $d\geqslant 3$. Using bounded $t$-structures, we construct $d$-hearts by using shifted Serre functors. This is a joint work in progress with Osamu Iyama.

A cluster algebra is a $\mathbb{Z}$-subalgebra of a rational function field generated by a special set of generators called cluster variables, which are grouped into overlapping subsets of fixed size, called clusters. One can travel from one cluster to the others by a recursive process called mutation. Cluster monomials are monomials in cluster variables from a single cluster. A fundamental problem in cluster algebras is to characterize when two cluster variables are contained in the same cluster. A more general problem is to characterize when the product of two cluster monomials is still a cluster monomial. In this talk, we introduce F-invariant in cluster algebras and use it to characterize when the product of two cluster monomials is still a cluster monomial.

It is believed that modular data, i.e., the set of S and T matrices, uniquely determine a modular tensor category. We prove that this conjecture is true for fusion category arising from quantum group $U_{q}(sl_{N})$. As a consequence, we prove a few Kazhdan-Lusztig type correspondences, that is, the representation categories of affine Lie algebras, W-algebras and quantum groups of type A are braided tensor equivalent.