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In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and describe some constructions of them. Then we study the formal distribution (noncommutative) Poisson algebras and coefficient (noncommutative) Poisson algebras. Next, we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations., 31 pages

Given a weight-one element $u$ of a vertex operator algebra $V$, we construct an automorphism of the category of generalized $g$-twisted modules for automorphisms $g$ of $V$ fixing $u$. We apply this construction to the case that $V$ is an affine vertex operator algebra to obtain explicit results on these automorphisms of categories. In particular, we give explicit constructions of certain generalized twisted modules from generalized twisted modules associated to diagram automorphisms of finite-dimensional simple Lie algebras and generalized (untwisted) modules., 32 pages

We give some general results about the generators and relations for the higher level Zhu algebras for a vertex operator algebra. In particular, for any element $u$ in a vertex operator algebra $V$, such that $u$ has weight greater than or equal to $-n$ for $n \in \mathbb{N}$, we prove a recursion relation in the $n$th level Zhu algebra $A_n(V)$ and give a closed formula for this relation. We use this and other properties of $A_n(V)$ to reduce the modes of $u$ that appear in the generators for $A_n(V)$ as long as $u \in V$ has certain properties (properties that apply, for instance, to the conformal vector for any vertex operator algebra or if $u$ generates a Heisenberg vertex subalgebra), and we then prove further relations in $A_n(V)$ involving such an element $u$. We present general techniques that can be applied once a set of reasonable generators is determined for $A_n(V)$ to aid in determining the relations of those generators, such as using the relations of those generators in the lower level Zhu algebras and the zero mode actions on $V$-modules induced from those lower level Zhu algebras. We prove that the condition that $(L(-1) + L(0))v$ acts as zero in $A_n(V)$ for $n \in \mathbb{Z}_+$ and for all $v$ in $V$ is a necessary added condition in the definition of the Zhu algebra at level higher than zero. We discuss how these results on generators and relations apply to the level $n$ Zhu algebras for the Heisenberg vertex operator algebra and the Virasoro vertex operator algebras at any level $n \in \mathbb{N}$., Comment: 38 pages; minor typos corrected. Published version

We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our first main result is a new shuffle realization of these highest weight modules and a combinatorial construction of their PBW and canonical bases in terms of Lyndon words. Our second main result is a classification of irreducible representations of orientifold KLR algebras and a computation of their global dimension in the case when the framing is trivial., 34 pages, some corrections in the proofs and improvements in the exposition

We consider an action of the automorphism group $\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$ on the filtered vector space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. This action induces on the associated graded vector space of $A_d(n)$, which is identified with the space $B_d(n)$ of open Jacobi diagrams, an action of the general linear group $\mathrm{GL}(n,Z)$ and an action of the graded Lie algebra of the IA-automorphism group of $F_n$ associated with its lower central series. We use these actions on $B_d(n)$ to study the $\mathrm{Aut}(F_n)$-module structure of $A_d(n)$. In particular, we consider the case where $d=2$ in detail and give an indecomposable decomposition of $A_2(n)$. We also construct a polynomial functor $A_d$ of degree $2d$ from the opposite category of the category of finitely generated free groups to the category of filtered vector spaces, which includes the $\mathrm{Aut}(F_n)$-module structure of $A_d(n)$ for all $n\geq 0$., 33 pages, some figures; a few sections are reorganized

The paper extends the notion of braided set and its close relative - the Yang-Baxter set - to the category of vector spaces and explore structure aspects of such a notion as morphisms and extensions. In this way we describe a family of solutions for the Yang-Baxter equation on the product of B and C if given B and C correspond to two linear (set-theoretic) solutions of the Yang-Baxter equation. One of the key observation is the relation of this question with the virtual pure braid group., Comment: 29 pages, 3 figures

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$. We give a polyhedral realization of the crystal basis for the extremal weight module of extremal weight $\lambda$, where $\lambda$ is an integral weight whose Weyl group orbit has neither a dominant integral weight nor an antidominant integral weight., Comment: 23 pages, 2 diagrams

Based on the NilHecke algebra $\mathsf{NH}_n$, the odd NilHecke algebra developed by Ellis, Khovanov and Lauda and Kang, Kashiwara and Tsuchioka's quiver Hecke superalgebra, we develop the Clifford Hecke superalgebra $\mathsf{NH}\mathfrak{C}_n$ as another super-algebraic analogue of $\mathsf{NH}_n$. We show that there is a notion of symmetric polynomials fitting in this picture, and we prove that these are generated by an appropriate analogue of elementary symmetric polynomials, whose properties we shall discuss in this text.

We continue the study of prime simple modules for quantum affine algebras from the perspective of $q$-fatorization graphs. In this paper we establish several properties related to simple modules whose $q$-factorization graphs are afforded by trees. The two most important of them are proved for type $A$. The first completes the classification of the prime simple modules with three $q$-factors by giving a precise criterion for the primality of a $3$-vertex line which is not totally ordered. Using a very special case of this criterion, we then show that a simple module whose $q$-factorization graph is afforded by an arbitrary tree is real. Indeed, the proof of the latter works for all types, provided the aforementioned special case is settled in general.

We give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over a pointed braided fusion category over an algebraically closed field., Minor corrections

We determine the level two Zhu algebra for the Heisenberg vertex operator algebra $V$ for any choice of conformal element. We do this using only the following information for $V$: the internal structure of $V$; the level one Zhu algebra of $V$ already determined by the second author, along with Vander Werf and Yang; and the information the lower level Zhu algebras give regarding irreducible modules. We are able to carry out this calculation of the level two Zhu algebra for $V$ with this minimal information by employing the general results and techniques for determining generators and relations for higher level Zhu algebras for a vertex operator algebra, as developed previously by the authors in "On generators and relations for higher level Zhu algebras and applications", by Addabbo and Barron, J. Algebra, 2023. In particular, we show that the level $n$ Zhu algebras for the Heisenberg vertex operator algebra become noncommutative at level $n=2$. We also give a conjecture for the structure of the level $n$ Zhu algebra for the Heisenberg vertex operator algebra, for any $n >2$., 65 pages; minor typos corrected; some calculations shortened

We categorify the commutation of Nakajima's Heisenberg operators $P_{\pm 1}$ and their infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot{gl_1})$ acting on the Grothendieck groups of Hilbert schemes. By combining our result with arxiv:1804.03645 , one obtains a geometric categorical $U_{q_1,q_2}(\ddot{gl_1})$ action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semi-divisorial log terminal singularities., Comment: V2:typos fixed. 29 pages

Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [4], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [9], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Green's formula., 25 pages. Published in Journal of Algebra (2023)

We study the holomorphic twist of 3d ${\cal N}=2$ gauge theories in the presence of boundaries, and the algebraic structure of bulk and boundary local operators. In the holomorphic twist, both bulk and boundary local operators form chiral algebras (\emph{a.k.a.} vertex operator algebras). The bulk algebra is commutative, endowed with a shifted Poisson bracket and a "higher" stress tensor; while the boundary algebra is a module for the bulk, may not be commutative, and may or may not have a stress tensor. We explicitly construct bulk and boundary algebras for free theories and Landau-Ginzburg models. We construct boundary algebras for gauge theories with matter and/or Chern-Simons couplings, leaving a full description of bulk algebras to future work. We briefly discuss the presence of higher A-infinity like structures., 96 pages

In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $\mathbb{A}$ in a modular fusion category $\mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $\mathbb{A}$ in $\mathcal{C}$, we introduce a ribbon category $\mathcal{C}_{\mathbb{A}}$ and show that it is again a modular fusion category. The definition of $\mathcal{C}_{\mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $\mathbb{A}$ is given by a simple commutative $\Delta$-separable Frobenius algebra $A$ in $\mathcal{C}$; (ii) when $\mathbb{A}$ is an orbifold datum in $\mathcal{C} = \operatorname{Vect}$, built from a spherical fusion category $\mathcal{S}$. We show that in case (i), $\mathcal{C}_{\mathbb{A}}$ is ribbon-equivalent to the category of local modules of $A$, and in case (ii), to the Drinfeld centre of $\mathcal{S}$. The category $\mathcal{C}_{\mathbb{A}}$ thus unifies these two constructions into a single algebraic setting., Comment: 58 pages

We describe new graphical models of the framed little disks operads which exhibit large symmetry dg Lie algebras.

For $$q \in {\mathbb {R}}$$ q ∈ R , $$|q| < 1$$ | q | < 1 we consider the universal enveloping $$C^*$$ C ∗ -algebra of a $$*$$ ∗ -algebra of q-canonical commutation relations (q-CCR), which is generated by $$a_1, \ldots , a_n$$ a 1 , … , a n subject to the relations $$\begin{aligned} a_i^* a_j = \delta _{ij} 1 + q a_j a_i^* . \end{aligned}$$ a i ∗ a j = δ ij 1 + q a j a i ∗ . It has a distinguished representation $$\pi _F$$ π F called the Fock representation, which is believed to be faithful. In this article we denote the image of the universal enveloping $$C^*$$ C ∗ -algebra of q-CCR in the Fock representation by $$\beth _{n,q}$$ ℶ n , q . The question whether $$C^*$$ C ∗ -isomorphism $$\beth _{n,q} \simeq \beth _{n,0}$$ ℶ n , q ≃ ℶ n , 0 holds has been considered in the literature and proved for $$|q| < 0.44$$ | q | < 0.44 . In this article we show that $$\beth _{n,q} \simeq \beth _{n,0}$$ ℶ n , q ≃ ℶ n , 0 for $$|q| < 1$$ | q | < 1 .

In this paper, we generalize Schur-Weyl duality and Morita Theorem on associative algebras to those on associative $H$-pseudoalgebras. Meanwhile, we get a plenty of associative $H$-pseudoalgebras over a cocommutative Hopf algebra $H$., All comments are welcome

We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on $\mathbb R^d$, generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of $\hbar$, giving strict quantizations on the space of analytic functions on $\mathbb R^d$ with infinite radius of convergence. We also address further questions such as continuity of the classical limit $\hbar \to 0$, compatibility with *-involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as *-algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example., 45 pages, final version

Using M-theory in physics, Cho, Gang, and Kim (JHEP 2020, 115 (2020) ) recently outlined a program that connects two parallel subjects of three dimensional manifolds, namely, geometric topology and quantum topology. They suggest that classical topological invariants such as Chern-Simons invariants of $\text{SL}(2,\mathbb{C})$-flat connections and adjoint Reidemeister torsions of a three manifold can be packaged together to produce a $(2+1)$-topological quantum field theory, which is essentially equivalent to a modular tensor category. It is further conjectured that every modular tensor category can be obtained from a three manifold and a semi-simple Lie group. In this paper, we study this program mathematically, and provide strong support for the feasibility of such a program. The program produces an algorithm to generate the potential modular $T$-matrix and the quantum dimensions of a candidate modular data. The modular $S$-matrix follows from essentially a trial-and-error procedure. We find modular tensor categories that realize candidate modular data constructed from Seifert fibered spaces and torus bundles over the circle that reveal many subtleties in the program. We make a number of improvements to the program based on our computations. Our main result is a mathematical construction of a premodular category from each Seifert fibered space with three singular fibers and a family of torus bundles over the circle with Thurston SOL geometry. The premodular categories from Seifert fibered spaces are related to Temperley-Lieb-Jones categories and the ones from torus bundles over the circle are related to metaplectic categories. We conjecture that a resulting premodular category is modular if and only if the three manifold is a $\mathbb{Z}_2$-homology sphere and condensation of bosons in premodular categories leads to either modular or super-modular categories., Comment: To appear in Comm. Math. Phys

A braided monoidal category may be considered a $3$-category with one object and one $1$-morphism. In this paper, we show that, more generally, $3$-categories with one object and $1$-morphisms given by elements of a group $G$ correspond to $G$-crossed braided categories, certain mathematical structures which have emerged as important invariants of low-dimensional quantum field theories. More precisely, we show that the 4-category of $3$-categories $\mathcal{C}$ equipped with a 3-functor $\mathrm{B}G \to \mathcal{C}$ which is essentially surjective on objects and $1$-morphisms is equivalent to the $2$-category of $G$-crossed braided categories. This provides a uniform approach to various constructions of $G$-crossed braided categories., Comment: 66 pages, many figures

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. We review some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. We also review the related analytic extension results, conjectures and problems. We discuss the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of $q$-traces and pseudo-$q$-traces of products of intertwining operators. We also discuss the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. We then explain conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras., Comment: 27 pages. To appear in a special issue of Chinese Annals of Mathematics, series B, in memory of Gu Chaohao

Recently, Nakaoka and Palu introduced a notion of extriangulated categories. This is a unification of exact categories and triangulated categories. In this paper, we generalize the definitions of Hall algebras of exact categories and triangulated categories to extriangulated categories., 21 pages

In this paper we mainly study the vertex operator algebra $\mathbb{C} \mathrm{VA}(e, f)$ generated by two Ising vectors $e$ and $f$ with $\langle e, f\rangle=\frac{5}{2^{10}}$. We prove $\mathbb{C} \mathrm{VA}(e, f)$ is isomorphic to the $6A$-algebra $\mathcal{U}_{6A}$ constructed by Lam-Yamada-Yamauchi. We also discuss the cases $\langle e, f\rangle=\frac{1}{4}, \frac{1}{2^{5}}, 0 $ and the case $\langle e, f\rangle=\frac{1}{2^{8}}$ with $(\tau_{e}\tau_{f})^4=1$.

Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebrasWe introduce the notion of anti-pre-Lie algebras as the underlying algebraic structures of nondegenerate commutative 2-cocycles which are the "symmetric" version of symplectic forms on Lie algebras. They can be characterized as a class of Lie-admissible algebras whose negative left multiplication operators make representations of the commutator Lie algebras. We observe that there is a clear analogy between anti-pre-Lie algebras and pre-Lie algebras by comparing them in terms of several aspects. Furthermore, it is unexpected that a subclass of anti-pre-Lie algebras, namely admissible Novikov algebras, correspond to Novikov algebras in terms of $q$-algebras. Consequently, there is a construction of admissible Novikov algebras from commutative associative algebras with derivations or more generally, admissible pairs. The correspondence extends to the level of Poisson type structures, leading to the introduction of the notions of anti-pre-Lie Poisson algebras and admissible Novikov-Poisson algebras, whereas the latter correspond to Novikov-Poisson algebras., 29 pages

There is a one-to-one correspondence between strong inversions on knots in the three-sphere and a special class of four-ended tangles. We compute the reduced Khovanov homology of such tangles for all strong inversions on knots with up to 9 crossings, and discuss these computations in the context of earlier work by the second author. In particular, we provide a counterexample to [Conjecture 29, arXiv:1311.1085] as well as a refinement of and additional evidence for [Conjecture 28, arXiv:1311.1085]., 15 pages, 8 colour figures

We apply the sheaf cohomology BGG method developed by the authors and Lachowska-Qi to the computation of Hochschild cohomology groups of various blocks of the small quantum group. All our computations of the center of the corresponding block agree with the conjectures of Lachowska-Qi. In the case of the nontrivial singular block for $\mathfrak{g} = \mathfrak{sl}_3$, we obtain the H*$(\mathfrak{u}, \mathbb{C}) = \mathbb{C}[\mathcal{N}]$-module structure of HH*$(\mathfrak{u}_{\lambda})$., Comment: 25 pages, 34 figures

In this paper, it is shown that the diagonal coset vertex operator algebra $C(L_{\mathfrak{g}}(k+2,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(2,0))$ is rational and $C_2$-cofinite in case $\mathfrak{g}=so(2n), n\geq 3$ and $k$ is an admissible number for $\hat{\mathfrak{g}}$. It is also shown that the diagonal coset vertex operator algebra $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ is rational and $C_2$-cofinite in case $k$ is an admissible number for $\hat{sl_2}$. Furthermore, irreducible modules of $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ are classified in case $k$ is a positive odd integer., 28 pages

In this paper we propose a reduction scheme for multivector fields phrased in terms of $L_\infty$-morphisms. Using well-know geometric properties of the reduced manifolds we perform a Taylor expansion of multivector fields, which allows us to built up a suitable deformation retract of DGLA's. We first obtained an explicit formula for the $L_\infty$-Projection and -Inclusion of generic DGLA retracts. We then applied this formula to the deformation retract that we constructed in the case of multivector fields on reduced manifolds. This allows us to obtain the desired reduction $L_\infty$-morphism. Finally, we perfom a comparison with other reduction procedures., 30 pages, comments are welcome!

We show that four-dimensional superconformal algebras admit an infinite-dimensional derived enhancement after performing a holomorphic twist. The type of higher symmetry algebras we find are closely related to algebras studied by Faonte-Hennion-Kapranov, Hennion-Kapranov, and the second author with Gwilliam in the context of holomorphic QFT. We show that these algebras are related to the two-dimensional chiral algebras extracted from four-dimensional superconformal theories by Beem and collaborators; further deforming by a superconformal element induces the Koszul resolution of a plane in $\mathbb{C}^2 \cong \mathbb{R}^4$. The central charges at the level of chiral algebras arise from central extensions of the higher symmetry algebras., 59 pages, two figures. v. 3: major edits to prepare for journal submission

We address the study of multiparameter quamtum groups (=MpQG's) at roots of unity, namely quantum universal enveloping algebras $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) $ depending on a matrix of parameters $ \boldsymbol{\rm q} = {\big( q_{ij} \big)}_{i, j \in I} \, $. This is performed via the construction of quantum root vectors and suitable "integral forms" of $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) \, $, a \textsl{restricted one} - generated by quantum divided powers and quantum binomial coefficients - and an \textsl{unrestricted\/} one - where quantum root vectors are suitably renormalized. The specializations at roots of unity of either forms are the "MpQG's at roots of unity" we look for. In particular, we study special subalgebras and quotients of our MpQG's at roots of unity - namely, the multiparameter version of small quantum groups - and suitable associated quantum Frobenius morphisms, that link the MpQG's at roots of 1 with MpQG's at 1, the latter being classical Hopf algebras bearing a well precise Poisson-geometrical content. A key point in the discussion - often at the core of our strategy - is that every MpQG is actually a 2-cocycle deformation of the algebra structure of (a lift of) the "canonical" one-parameter quantum group by Jimbo-Lusztig, so that we can often rely on already established results available for the latter. On the other hand, depending on the chosen multiparameter $ \boldsymbol{\rm q} $ our quantum groups yield (through the choice of integral forms and their specialization) different semiclassical structures, namely different Lie coalgebra structures and Poisson structures on the Lie algebra and algebraic group underlying the canonical one-parameter quantum group., 71 pages. Final version, shortened and polished after the referee's comments - the mathematical content, though, is unchanged with respect to the previous version. To appear in "Journal of Noncommutative Geometry"

For any triple $(i,a,\mu)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $\mu$, we define a quiver current $H^{(i,a)}_\mu$ acting on the tensor power $\Lambda^Q$ of symmetric functions over the vertices of $Q$. These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple $(\mathbf{i},\mathbf{a},\mu(\bullet))$ of sequences of such data, we define the quiver Hall-Littlewood function $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ as the result of acting on $1\in\Lambda^Q$ by the corresponding sequence of quiver currents. The quiver Kostka-Shoji polynomials are the expansion coefficients of $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ in the tensor Schur basis. These polynomials include the Kostka-Foulkes polynomials and parabolic Kostka polynomials (Jordan quiver) and the Kostka-Shoji polynomials (cyclic quiver) as special cases. We show that the quiver Kostka-Shoji polynomials are graded multiplicities in the equivariant Euler characteristic of a vector bundle on Lusztig's convolution diagram determined by the sequences $\mathbf{i},\mathbf{a}$. For certain compositions of currents we conjecture higher cohomology vanishing of the associated vector bundle on Lusztig's convolution diagram. For quivers with no branching we propose an explicit positive formula for the quiver Kostka-Shoji polynomials in terms of catabolizable multitableaux. We also relate our constructions to $K$-theoretic Hall algebras, by realizing the quiver Kostka-Shoji polynomials as natural structure constants and showing that the quiver currents provide a symmetric function lifting of the corresponding shuffle product. In the case of a cyclic quiver, we explain how the quiver currents arise in Saito's vertex representation of the quantum toroidal algebra of type $\mathfrak{sl}_r$.

We study surface defects in three-dimensional topological quantum field theories which separate different theories of Reshetikhin-Turaev type. Based on the new notion of a Frobenius algebra over two commutative Frobenius algebras, we present an explicit and computable construction of such defects. It specialises to the construction in Carqueville et al., arXiv:1710.10214 if all 3-strata are labelled by the same topological field theory. We compare the results to the model-independent analysis in Fuchs et al., arXiv:1203.4568 and find agreement., 42 pages

For a fixed positive integer $k$, any element $g$ of the permutation group $S_{k}$ acts on the tensor product vertex operator algebra $V^{\otimes k}$ in the obvious way. In this paper, we determine the $S$-matrix of $\left(V^{\otimes k}\right)^{G}$ if $G=\left\langle g\right\rangle $ is the cyclic group generated by $g=\left(1,\ 2,\cdots,k\right).$, Comment: 20 pages

This is a survey on recent progress in algebraic deformation theory from the point of view of Koszul duality of algebraic operads., 95 pages

Let $V(1)$ be the natural representation of $U(\mathfrak{sl}_2).$ The multiplicities of $V(k)$ in $V(1)^{\otimes N}$ have multiple interpretations in combinatorics. In this paper, we investigate one such combinatorial interpretation of their multiplicities. Furthermore, we extend our analysis to the two-dimensional representation $T(1)$ of the restricted quantum enveloping algebra $U_q^{\mathrm{res}}(\mathfrak{sl}_2)$. By employing combinatorial techniques, this study aims to elucidate the multiplicity of $T(k)$ within $T(1)^{\otimes N}$ while also offering a comprehensive analysis of the asymptotic behavior of these multiplicities as the parameter $N$ approaches infinity., 18 pages

The well-known Loday-Quillen-Tsygan theorem calculates the Lie algebra homology of the infinite general linear Lie algebra $\mathfrak{gl}(A)$ over an unital associative algebra $A$. We generalize the Loday-Quillen-Tsygan theorem to an infinite Lie algebra associated with a (framed) quiver, where we assign to each vertex $v$ an infinite general linear Lie algebra $\mathfrak{gl}(A_v)$, to each edge $e$ an infinite matrix module and to each framed vertex a (anti)-fundamental representation. Given this data, each loop or path ending on framed vertices of the quiver defined a stratified factorization algebra over $S^1$ or $[0,1]$ respectively. We show that the corresponding Lie algebra homology can be expressed as summing the factorization homology over all loops and framed paths of the quiver.

For a simple, simply connected, complex group G, we prove an explicit formula to compute the Atiyah class of parabolic determinant of cohomology line bundle on the moduli space of parabolic $G$-bundles. This generalizes an earlier result of Beilinson-Schechtman., Split from arXiv:2103.03792 as a separate paper based on suggestions of a referee with some additional details

A family of heptagon relations -- algebraic imitations of five-dimensional Pachner moves -- is constructed, parameterized by simplicial 3-cocycles. These relations are shown to possess a nontrivial "heptagon cohomology", which yields invariants of a pair "manifold, its 3rd cohomology class"., 23 pages

In 2015, Lusztig [Bull. Inst. Math. Acad. Sin. (N.S.)10(2015), no.1, 1-72] showed that for a connected reductive group over an algebraic closure of a finite field the associated (geometric) Hecke category admits a truncation in a two-sided Kazhdan--Lusztig cell, making it a categorification of the asymptotic algebra (J-ring), and that the categorical center of this "asymptotic Hecke category" is equivalent to the category of unipotent character sheaves supported in the cell. Subsequently, Lusztig noted that an asymptotic Hecke category can be constructed for any finite Coxeter group using Soergel bimodules. Lusztig conjectured that the centers of these categories are modular tensor categories (which was then proven by Elias and Williamson) and that for non-crystallographic finite Coxeter groups the S-matrices coincide with the Fourier matrices that were constructed in the 1990s by Lusztig, Malle, and Brou\'e--Malle. If the conjecture is true, the centers may be considered as categories of "unipotent character sheaves" for non-crystallographic finite Coxeter groups. In this paper, we show that the conjecture is true for dihedral groups and for some (we cannot resolve all) cells of H3 and H4. The key ingredient is the method of H-reduction and the identification of the (reduced) asymptotic Hecke category with known categories whose center is already known as well. We conclude by studying the asymptotic Hecke category and its center for some infinite Coxeter groups with a finite cell.

This paper studies planar algebras of Jones' style associated with the Young graph. We first see that, every time we fix a harmonic function on the Young graph, we may obtain a planar algebra whose structure is defined in terms of a state sum over the ways of filling planar tangles with Young diagrams. We delve into the case that the harmonic function is related to the Plancherel measures on Young diagrams. Along with an element that is depicted as a cross of two strings, we see that the defining relations among morphisms for the Khovanov Heisenberg category are recovered in the planar algebra. We also identify certain elements in the planar algebra with particular functions of Young diagrams that include the moments, Boolean cumulants and normalized characters. This paper thereby bridges diagramatical categorification and asymptotic representation theory. In fact, the Khovanov Heisenberg category is one of the most fundamental examples of diagramatical categorification whereas the harmonic functions on the Young graph have been a central object in the asymptotic representation theory of symmetric groups., 51pages

We establish a q-version of the Schur-Weyl duality, in which the role of the symmetric group is played by the Hecke algebra and the role of the enveloping algebra U(gl(N)) is played by the Reflection Equation algebra, associated with any skew-invertible Hecke symmetry. Also, in each Reflection Equation algebra we define analogues of the Schur polynomials and power sums in two forms: as polynomials in generators of a given Reflection Equation algebra and in terms of the so-called eigenvalues of the generating matrix $L$, defined by means of the Cayley-Hamilton identity. It is shown that on any Reflection Equation algebra there exists a formula, which brings into correlation the Schur polynomials and power sums by means of the characters of the Hecke algebras in the spirit of the classical Frobenius formula.

We consider reduced imaginary Verma modules for the untwisted quantum affine algebras $U_q(\hat{\g})$ and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra $\mathcal K_q$ constructed in earlier work by Ben Cox and two of the authors. We prove the existence of the imaginary crystal base for any object in a suitable category $\mc{O}^q_{red,im}$ containing the reduced imaginary Verma modules for $U_q(\hat{\g})$., 26 pages

We present resent results regarding involutive, non-degenerate solutions of the set-theoretic Yang-Baxter and reflection equations. We recall the notion of braces and we present and prove various fundamental properties necessary for the solution of the set theoretic Yang-Baxter equation. We then consider lambda parametric set-theoretic solutions of the Yang-Baxter equation and we extract the associated quantum algebra. We also discuss the notion of the Drinfeld twist for involutive solutions and their relation to the Yangian. We then focus on reflections and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the symmetric group. We show that there exists a ``reflection'' finite sub-algebra for some special choice of reflection maps., 17 pages, LaTex. arXiv admin note: text overlap with arXiv:2003.08317

We apply the cobordism hypothesis with singularities to the case of affine Rozansky--Witten models, providing a construction of extended TQFTs that includes all line and surface defects. On a technical level, this amounts to proving that the associated homotopy 2-category is pivotal, and to systematically employing its 3-dimensional graphical calculus. This in particular allows us to explicitly calculate state spaces for surfaces with arbitrary defect networks. As specific examples we discuss symmetry defects which can be used to model non-trivial background gauge fields, as well as boundary conditions., 79 pages

Let $\mathcal{A}$ be a generalized Weyl or $q$-Weyl algebra, $M$ be an $\mathcal{A}$-$\overline{\mathcal{A}}$ bimodule. Choosing an automorphism $\rho$ of $\mathcal{A}$ we can define the notion of an invariant Hermitian form: $(au,v)=(u,v\rho(a))$ for all $a\in \mathcal{A}$ and $u,v\in M$. Papers [P. Etingof, D. Klyuev, E. Rains, D. Stryker. Twisted traces and positive forms on quantized Kleinian singularities of type A. SIGMA 17 (2021), 029, 31 pages, arXiv:2009.09437] and [D. Klyuev. Twisted traces and positive forms on generalized q-Weyl algebras. SIGMA 18 (2022), 009, 28 pages, arXiv:2105.12652] obtained a classification of invariant positive definite Hermitian forms in the case when $M=\mathcal{A}=\overline{\mathcal{A}}$, the case of the regular bimodule. We obtain a classification of invariant positive definite forms on $M$ in the case when $\mathcal{A}$ has no finite-dimensional representations., Comment: 42 pages

We show that the canonical equivalences of categories between 2-dimensional (unoriented) topological quantum field theories valued in a symmetric monoidal category and (extended) commutative Frobenius algebras in that symmetric monoidal category are symmetric monoidal equivalences. As an application, we recover that the invariant of 2-dimensional manifolds given by the product of (extended) commutative Frobenius algebras in a symmetric tensor category is the multiplication of the invariants given by each of the algebras., 4 pages

The quandle coloring quiver was introduced by Cho and Nelson as a categorification of the quandle coloring number. In some cases, it has been shown that the quiver invariant offers more information than other quandle enhancements. In this paper, we compute the quandle coloring quivers of 2-bridge links with respect to the dihedral quandles.

This paper studies the universal first-order Massey product of a prefactorization algebra, which encodes higher algebraic operations on the cohomology. Explicit computations of these structures are carried out in the locally constant case, with applications to factorization envelopes on $\mathbb{R}^m$ and a compactification of linear Chern-Simons theory on $\mathbb{R}^2\times \mathbb{S}^1$., 41 pages

We define a notion of nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JBW}^*$-algebra (Jordan von Neumann algebra) equipped with a normal faithful state $\varphi$. In the particular case of $\mathrm{JW}^*$-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative $\mathrm{L}^p$-spaces. We also make the link with the nonassociative $\mathrm{L}^p$-spaces of Iochum associated to $\mathrm{JBW}$-algebras and the investigation of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. More precisely, we show that our nonassociative $\mathrm{L}^p$-spaces contain isometrically the $\mathrm{L}^p$-spaces of Iochum and that all tracial nonassociative $\mathrm{L}^p$-spaces from $\mathrm{JW}^*$-factors arise as positively contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces., Comment: 24 pages, contains in particular some sections of an old and too long version of arXiv:1909.00391