We give an interpretation of the $(q,t)$-deformed Cartan matrices of finite type and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiss-Leclerc-Schr{o}er~[Invent.~math.
~{bf{209}} (2017)]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez-Leclerc~[J.~Eur.~Math.~Soc.~{bf 18} (2016)], and propose a conjecture that their dimensions coincide with the pole orders of the normalized $R$-matrices between the corresponding Kirillov-Reshetikhin modules.
We study the deformation complex of the dg wheeled properad of $mathbb{Z}$-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck-Teichmu
ller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application we classify all universal quantizations of $mathbb{Z}$-graded quadratic Poisson structures together with the underlying (so called) homogeneous formality maps.
We introduce a subalgebra $overline F$ of the Clifford vertex superalgebra ($bc$ system) which is completely reducible as a $L^{Vir} (-2,0)$-module, $C_2$-cofinite, but it is not conformal and it is not isomorphic to the symplectic fermion algebra $m
athcal{SF}(1)$. We show that $mathcal{SF}(1)$ and $overline{F}$ are in an interesting duality, since $overline{F}$ can be equipped with the structure of a $mathcal{SF}(1)$-module and vice versa. Using the decomposition of $overline F$ and a free-field realization from arXiv:1711.11342, we decompose $L_k(mathfrak{osp}(1vert 2))$ at the critical level $k=-3/2$ as a module for $L_k(mathfrak{sl}(2))$. The decomposition of $L_k(mathfrak{osp}(1vert 2))$ is exactly the same as of the $N=4$ superconformal vertex algebra with central charge $c=-9$, denoted by $mathcal V^{(2)}$. Using the duality between $overline{F}$ and $mathcal{SF}(1)$, we prove that $L_k(mathfrak{osp}(1vert 2))$ and $mathcal V^{(2)}$ are in the duality of the same type. As an application, we construct and classify all irreducible $L_k(mathfrak{osp}(1vert 2))$-modules in the category $mathcal O$ and the category $mathcal R$ which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra $N_{-3/2}(mathfrak{osp}(1vert 2))$ as a $N_{-3/2}(mathfrak{sl}(2))$-module. We extend this example, and for each $p ge 2$, we introduce a non-conformal vertex algebra $mathcal A^{(p)}_{new}$ and show that $mathcal A^{(p)}_{new} $ is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra $ mathcal V^{(p)} _{new}$ which is isomorphic to the logarithmic vertex algebra $mathcal V^{(p)}$ as a module for $widehat{mathfrak{sl}}(2)$.
This paper is devoted to the study of Keller admissible triples. We prove that a Keller admissible triple induces an isomorphism of Gerstenhaber algebras between Hochschild cohomologies of the direct-sum type for dg algebras. As an application, we gi
ve an alternative proof of the Kontsevich-Duflo theorem for finite-dimensional Lie algebras.
Biquandle brackets define invariants of classical and virtual knots and links using skein invariants of biquandle-colored knots and links. Biquandle coloring quivers categorify the biquandle counting invariant in the sense of defining quiver-valued e
nhancements which decategorify to the counting invariant. In this paper we unite the two ideas to define biquandle bracket quivers, providing new categorifications of biquandle brackets. In particular, our construction provides an infinite family of categorifications of the Jones polynomial and other classical skein invariants.
We specialise the construction of orbifold graph TQFTs introduced in Carqueville et al., arXiv:2101.02482 to Reshetikhin-Turaev defect TQFTs. We explain that the modular fusion category ${mathcal{C}}_{mathcal{A}}$ constructed in Muleviv{c}ius-Runkel,
arXiv:2002.00663 from an orbifold datum $mathcal{A}$ in a given modular fusion category $mathcal{C}$ is a special case of the Wilson line ribbon categories introduced as part of the general theory of orbifold graph TQFTs. Using this, we prove that the Reshetikhin-Turaev TQFT obtained from ${mathcal{C}}_{mathcal{A}}$ is equivalent to the orbifold of the TQFT for $mathcal{C}$ with respect to the orbifold datum $mathcal{A}$.
Given an orthogonal compact matrix quantum group defined by intertwiner relations, we characterize by relations its projective version. As a sample application, we prove that $PU_n^+=PO_n^+$. We also give a combinatorial proof of the fact that $S_{n^2}^+$ is monoidally equivalent to $PO_n^+$.
This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modula
r extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara-Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generalize to braided fusion categories with the fusion rules of the representation categories of extraspecial $p$-groups for any prime $p$, which possess a minimal modular extension only if they arise as the adjoint subcategory of a twisted double of an extraspecial $p$-group.
Let $G$ be a connected complex semi-simple Lie group and ${mathcal{B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${mathcal{B}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, wh
ich contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.
We give a proof of Lusztigs conjectural multiplicity formula for non-restricted modules over the De Concini-Kac type quantized enveloping algebra at $ell$-th root of unity, where $ell$ is an odd prime power satisfying certain reasonable conditions.
