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We introduce and study a family of power series, which we call Theta series, whose coefficients are in the tensor square of a quantum loop algebra. They arise from a coproduct factorization of the T-series of Frenkel--Hernandez, which are leading terms of transfer matrices of certain infinite-dimensional irreducible modules over the upper Borel subalgebra in the category O of Hernandez--Jimbo. We prove that each weight component of a Theta series is polynomial. As applications, we establish a decomposition formula and a polynomiality result for R-matrices between an irreducible module and a finite-dimensional irreducible module in category O. We extend T-series and Theta series to Yangians by solving difference equations determined by the truncation series of Gerasimov--Kharchev--Lebedev--Oblezin. We prove polynomiality of Theta series by interpreting them as associators for triple tensor product modules over shifted Yangians., 65 pages, comments welcome

In this note we review a construction of a BPS Hilbert space in an effective supersymmetric quiver theory with 4 supercharges. We argue abstractly that this space contains elements of an equivariant generalized cohomology theory $E_G^{*}(-)$ of the quiver representation moduli space giving concretely Dolbeault cohomology, K-theory or elliptic cohomology depending on the spacial slice is compactified to a point, a circle or a torus respectively, and something more amorphous in other cases. Furthermore BPS instantons -- basic contributors to interface defects or a Berry connection -- induce a BPS algebra on the BPS Hilbert spaces representing Fourier-Mukai transforms on the quiver representation moduli spaces descending to an algebra over $E_G^{*}(-)$ as its representation. In the cases when the quiver describes a toric Calabi-Yau three-fold (CY${}_3$) the algebra is a respective generalization of the quiver BPS Yangian algebra discussed in the literature, in more general cases it is given by an abstract generalized cohomological Hall algebra., Comment: 31 pages, 2 figures, minor corrections

In this paper we introduce an analog of the (classical) Yang-Baxter equation (CYBE) for vertex operator algebras (VOAs) in its tensor form, called the vertex operator Yang-Baxter equation (VOYBE). When specialized to level one of a vertex operator algebra, the VOYBE reduces to the CYBE for Lie algebras. To give an operator form of the VOYBE, we also introduce the notion of relative Rota-Baxter operators (RBOs) as the VOA analog of relative RBOs (classically called $\mathcal{O}$-operators) for Lie algebras. It is shown that skewsymmetric solutions $r$ to the VOYBE in a VOA $U$ are characterized by the condition that their corresponding linear maps $T_r:U'\to U$ from the graded dual $U'$ of $U$ are relative RBOs. On the other hand, strong relative RBOs on a VOA $V$ associated to an ordinary $V$-module $W$ are characterized by the condition that their antisymmetrizers are solutions to the $0$-VOYBE in the semidirect product VOA $V\rtimes W'$. Specializing to the first level of a VOA, these relations between the solutions of the VOYBE and the relative RBOs for VOAs recover the classical relations between the solutions of the CYBE and the relative RBOs for Lie algebras., 30 pages

These are notes from talks given at a spring school on topological quantum field theory in Nova Scotia during May of 2023. The aim is to introduce the reader to the role of factorization algebras and related concepts in field theory. In particular, we discuss the relationship between factorization algebras, $\mathbb{E}_n$-algebras, vertex algebras, and the functorial perspective on field theories., 42 pages. Comments welcome!

We demonstrate that commutativity of numerous one-dimensional subalgebras in $W_{1+\infty}$ algebra, i.e. the existence of many non-trivial integrable systems described in recent arXiv:2303.05273 follows from the subset of relations in algebra known as Serre relations. No other relations are needed for commutativity. The Serre relations survive the deformation to the affine Yangian $Y(\hat{\mathfrak{gl}}_1)$, hence the commutative subalgebras do as well. A special case of the Yangian parameters corresponds to the $\beta$-deformation. The preservation of Serre relations can be thought of a selection rule for proper systems of commuting $\beta$-deformed Hamiltonians. On the contrary, commutativity in the extended family associated with ``rational (non-integer) rays" is {\it not} reduced to the Serre relations, and uses also other relations in the $W_{1+\infty}$ algebra. Thus their $\beta$-deformation is less straightforward., Comment: 13 pages

In this paper, we associate the TKK algebra $\widehat{\mathcal{G}}(\mathcal{J})$ with vertex algebras through twisted modules. Firstly, we prove that for any complex number $\ell$, the category of restricted $\widehat{\mathcal{G}}(\mathcal{J})$-modules of level $\ell$ is canonically isomorphic to the category of $\sigma$-twisted $V_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$-modules, where $V_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$ is a vertex algebra arising from the toroidal Lie algebra of type $C_2$ and $\sigma$ is an isomorphism of $V_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$ induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer $\ell$, the integrable restricted $\widehat{\mathcal{G}}(\mathcal{J})$-modules of level $\ell$ are exactly the $\sigma$-twisted modules for the quotient vertex algebra $L_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$ of $V_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$. Finally, we classify the irreducible $\frac{1}{2}{\mathbb{N}}$-graded $\sigma$-twisted $L_{\widehat{\mathcal{C}_{\mathfrak g}}}(\ell,0)$-modules., Comment: 19 pages

For a twisted $C^*$-dynamical system $(\mathscr{A},\mathbb{R}^n,\alpha,e)$ over a unital $C^*$-algebra we establish a weakly parametric pseudodifferential calculus analogously to the celebrated weakly parametric calculus due to Grubb and Seeley. If the $C^*$-algebra $\mathscr{A}$ has an $\alpha$-invariant trace then we prove an expansion of the resolvent trace (with respect to the dual trace on multipliers) for suitable pseudodifferential multipliers. The question whether the expansion holds true as a Hilbert space trace expansion in concrete GNS spaces for $\mathscr{A}$ will be addressed in a future publication., Comment: 27 pages

Motivated by the so-called H-cell reduction theorems, we investigate certain classes of bicategories which have only one H-cell apart from possibly the identity. We show that H_0-simple quasi fiab bicategories with unique H-cell H_0 are fusion categories. We further study two classes of non-semisimple quasi-fiab bicategories with a single H-cell apart from the identity. The first is $\cH_A$, indexed by a finite-dimensional radically graded basic Hopf algebra A, and the second is $\cG_A$, consisting of symmetric projective A-A-bimodules. We show that $\cH_A$ can be viewed as a 1-full subbicategory of $\cG_A$ and classify simple transitive birepresentations for $\cG_A$. We point out that the number of equivalence classes of the latter is finite, while that for $\cH_A$ is generally not., 28 pages

Various monoidal categories, including suitable representation categories of vertex operator algebras, admit natural Grothendieck-Verdier duality structures. We recall that such a Grothendieck-Verdier category comes with two tensor products which should be related by distributors obeying pentagon identities. We discuss in which circumstances these distributors are isomorphisms. This is achieved by taking the perspective of module categories over monoidal categories, using in particular the natural weak module functor structure of internal Homs and internal coHoms. As an illustration, we exhibit these concepts concretely in the case of categories of bimodules over associative algebras., 24 pages

We prove a recent conjecture of Manolescu-Willis which states that the $s$-invariant of a knot in $\mathbb{RP}^3$ (as defined by them) gives a lower bound on its null-homologous slice genus in the unit disk bundle of $TS^2$. We also conjecture a lower bound in the more general case where the slice surface is not necessarily null-homologous, and give its proof in some special cases., 8 pages

We study characters of states in $p$-adic vertex operator algebras. In particular, we show that the image of the character map for both the $p$-adic Heisenberg and $p$-adic lattice vertex operator algebras contain infinitely-many non-classical $p$-adic modular forms which are not contained in the image of the algebraic character map. We obtain also new expressions for square-bracket modes in the Heisenberg VOA which are used in the study of such characters., 15 pages

We define affine Super Yangian $Y_{\hbar}(\hat{sl}(m|n), \Pi) $ for affine special linear superalgebra $\hat{sl}(m|n)$ and arbitrary system of simple roots $\Pi$ in terms of minimalistic system of generators. We also consider Drinfeld presentation for affine super Yangian in the case of arbitrary simple root system $\Pi$ and prove that these two presentations (Drinfeld and minimalistic) of $Y_{\hbar}(\hat{sl}(m|n), \Pi)$ are isomorphic as associative superalgebras. We also construct isomorphism of affine super Yangians $Y_{\hbar}(\hat{sl}(m|n), \Pi)$ and $Y_{\hbar}(\hat{sl}(m|n), \Pi')$ for different simple root systems $\Pi$ and $\Pi'$. After them we also define Weyl groupoid as a set of morphisms in category with objects, which are super Yanginas $Y_{\hbar}(\hat{sl}(m|n), \Pi)$, where $\Pi$ is simple root system. We describe Weyl groupoid in terms of generators and describe action of these generators on super Yangians. We describe isomorphisms between $Y_{\hbar}(\hat{sl}(m|n), \Pi)$ and $Y_{\hbar}(\hat{sl}(m|n), \Pi')$ as elements of Weyl groupoid.

The derived moduli stack of complexes of vector bundles on a Gorenstein Calabi-Yau curve admits a 0-shifted Poisson structure. Feigin-Odesskii Poisson varieties are examples of such moduli spaces over complex elliptic curves. Using moduli stack of chains we construct an auxiliary Poisson varieties with a Poisson morphism from it to a Feigin-Odesskii variety. We call it the \emph{bosonization} of Feigin-Odesskii variety. As an application, we show that the Feigin-Odesskii Poisson brackets on projective spaces (associated with stable bundles of arbitrary rank on elliptic curves) admit no infinitesimal symmetries.

Originating in the work of A.M. Semikhatov and D. Adamovi\'c, inverse reductions are embeddings involving W-algebras corresponding to the same Lie algebra but different nilpotent orbits. Here, we show that an inverse reduction embedding between the affine $\mathfrak{sl}_{n+1}$ vertex operator algebra and the minimal $\mathfrak{sl}_{n+1}$ W-algebra exists. This generalises the realisations for $n=1,2$ in [arXiv:1711.11342, arXiv:2110.15203]. A similar argument is then used to show that inverse reduction embeddings exists between all hook-type $\mathfrak{sl}_{n+1}$ W-algebras, which includes the principal/regular, subregular, minimal $\mathfrak{sl}_{n+1}$ W-algebras, and the affine $\mathfrak{sl}_{n+1}$ vertex operator algebra. This generalises the regular-to-subregular inverse reduction of [arXiv:2111.05536], and similarly uses free-field realisations and their associated screening operators., Comment: 27 pages, 1 figure

The Drinfeld double associated to the weak multiplier Hopf ($*$-) algebra pairing $\left\langle A, B\right\rangle$ is constructed. We show that the Drinfeld double is again a weak multiplier Hopf ($*$-) algebra. If $A$ and $B$ are algebraic quantum groupoids, then so does the double. We also prove the correspondence between modules over the Drinfeld double and Yetter-Drinfeld modules. Finally, we prove that the double is a quasitriangular weak multiplier Hopf algebra.

We study the affine analogue $\mathrm{FT}_p(\mathfrak{sl}_2)$ of the triplet algebra. We show that $\mathrm{FT}_p(\mathfrak{sl}_2)$ is quasi-lisse and the associated variety is the nilpotent cone of $\mathfrak{sl}_2$. We realize $\mathrm{FT}_p(\mathfrak{sl}_2)$ as the global sections of a sheaf of vertex algebras in the spirit of Feigin--Tipunin and thereby construct infinitely many simple modules and, in particular solve a conjecture by Semikhatov and Tipunin. We introduce the Kazama--Suzuki dual superalgebra $s\mathcal{W}_p(\mathfrak{sl}_{2|1})$ of $\mathrm{FT}_p(\mathfrak{sl}_2)$ and their singlet type subalgebras $s\mathcal{M}_p(\mathfrak{sl}_{2|1})$ and $\mathcal{M}_p(\mathfrak{sl}_2)$ and show their correspondence of categories. For $p=1$, we show the logarithmic Kazhdan--Lusztig correspondence for these (super)algebras and, in particular, show that the quantum group corresponding to $s\mathcal{M}_1(\mathfrak{sl}_{2|1})$ is the unrolled restricted quantum supergroup $u^H_{-1}(\mathfrak{sl}_{2|1})$ as suggested by Semikhatov and Tipunin., 45 pages, comments are welcome!

By utilizing the technique introduced in our previous work to construct Hopf superalgebras by an inverse procedure of the Radford-Majid bosonization, we classify non-semisimple pointed Hopf superalgebras of dimension up to 10 over an algebraically closed field of characteristic zero., 26 pages

We prove the stability conjecture of $\imath$canonical bases, which was raised by Huanchen Bao and Weiqiang Wang in 2016, for all locally finite types. To this end, we characterize the trivial module over the $\imath$quantum groups of such type at $q = \infty$. This result can be seen as a very restrictive version of the $\imath$crystal base theory for locally finite types., 14 pages

In this article, we present some results and constructions about the Chiu-Tamarkin invariant motivated by the idempotence of microlocal kernels, including: (1) a natural explanation for the definition of the $\mathbb{Z}/\ell$-equivariant Chiu-Tamarkin invariant; (2) a graded commutative product on the non-equivariant Chiu-Tamarkin invariant; and (3) a construction of the $S^1$-equivariant Chiu-Tamarkin invariant. As applications, we: (1) construct a sequence of symplectic capacities $(\overline{c}_k)_{k\in \mathbb{N}}$ and prove that it coincides with the symplectic capacities $({c}_k)_{k\in \mathbb{N}}$ we defined using the $\mathbb{Z}/\ell$-equivariant Chiu-Tamarkin invariant under certain conditions; and (2) prove a Viterbo isomorphism. In the Appendix, we provide a proof of admissibility for all open sets in a cotangent bundle under the setup of triangulated categories., 56 pages. Comments are welcome!

We define the ordinary Minkowski space inside the conformal space according to Penrose and Manin as homogeneous spaces for the Poincar\'e and conformal group respectively. We realize the supersymmetric (SUSY) generalizations of such homogeneous spaces over the complex and the real fields. We finally investigate chiral (antichiral) superfields, which are superfields on the super Grassmannian, Gr(2|1, 4|1), respectively on Gr(2|0, 4|1). They ultimately give the twistor coordinates necessary to describe the conformal superspace as the flag Fl(2|0, 2|1; 4|1) and the Minkowski superspace as its big cell., Comment: 10 pages. To appear in the proceedings of the ' Avenues of Quantum Field Theory in Curved Spacetime September 14-16, 2022. University of Genova (Italy)

Let $\mathcal{U}$ be a braided tensor category, typically unknown, complicated and in particular non-semisimple. We characterize $\mathcal{U}$ under the assumption that there exists a commutative algebra $A$ in $\mathcal{U}$ with certain properties: Let $\mathcal{C}$ be the category of local $A$-modules in $\mathcal{U}$ and $\mathcal{B}$ the category of $A$-modules in $\mathcal{U}$, which are in our set-up usually much simpler categories than $\mathcal{U}$. Then we can characterize $\mathcal{U}$ as a relative Drinfeld center $\mathcal{Z}_\mathcal{C}(\mathcal{B})$ and $\mathcal{B}$ as representations of a certain Hopf algebra inside $\mathcal{C}$. In particular this allows us to reduce braided tensor equivalences to the knowledge of abelian equivalences, e.g. if we already know that $\mathcal{U}$ is abelian equivalent to the category of modules of some quantum group $U_q$ or some generalization thereof, and if $\mathcal{C}$ is braided equivalent to a category of graded vector spaces, and if $A$ has a certain form, then we already obtain a braided tensor equivalence between $\mathcal{U}$ and $\mathrm{Rep}(U_q)$. A main application of our theory is to prove logarithmic Kazhdan-Lusztig correspondences, that is, equivalences of braided tensor categories of representations of vertex algebras and of quantum groups. Here, the algebra $A$ and the corresponding category $\mathcal{C}$ are a-priori given by a free-field realization of the vertex algebra and by a Nichols algebra. We illustrate this in those examples where the representation theory of the vertex algebra is well enough understood. In particular we prove the conjectured correspondences between the singlet vertex algebra $\mathcal{M}(p)$ and the unrolled small quantum groups of $\mathfrak{sl}_2$ at $2p$-th root of unity. Another new example is the Kazhdan-Lusztig correspondence for $\mathfrak{gl}_{1|1}$., 92 pages

We continue the work begun by Mickler-Moll investigating the properties of the polynomial eigenfunctions of the Nazarov-Sklyanin quantum Lax operator. By considering products of these eigenfunctions, we produce a novel generalization of a formula of Kerov relating Jack Littlewood-Richardson coefficients and residues of certain rational functions. Precisely, we derive a system of constraints on Jack Littlewood-Richardson coefficients in terms of a simple multiplication operation on partitions., 62 pages, 7 figures

We construct an $\mathfrak{sl}_2$-action on equivariant $\mathfrak{gl}_N$-link homologies. As a consequence we obtain an action of $\mathfrak{sl}_2$ on these homologies as well as a $p$-DG structures for $p$ a prime number. We explore topological applications of these structures., 53 pages, many figures. Comments welcome! 2nd version removes a mistakes in the topological applications of the construction

We establish precise counts of cubic $\mathbb{F}_q[t]$-algebras and quartic $\mathbb{F}_q[t]$-algebras with cubic resolvent of discriminant $q^b$. To get these counts, the main geometric input is our construction of smooth compactifications of simply branched degree $d$ Hurwitz stacks in mixed characteristic for $d \in \{3,4,5\}$ as stacks of orbifold quasimaps to $B\mathbf{S}_d$, which is presented as a GIT quotient stack $[V_d^{ss}/G_d]$ for a prehomogeneous vector space $(G_d,V_d)$. The main number-theoretic input is Igusa's computations of the Igusa zeta functions for $(G_3,V_3)$ and $(G_4,V_4)$. Our compactifications of Hurwitz stacks of a curve $C$ admit proper extensions of the branch map to $\mathrm{Sym}^n(C)$. Because these extensions are small, we can compute the intersection cohomology of certain local systems on $\mathrm{Conf}^n(C) \subset \mathrm{Sym}^n(C)$ as the cohomology of our compactifications, answering questions of Ellenberg-Tran-Westerland and Kapranov-Schechtman. Using a comparison theorem of Kapranov-Schechtman, we compute the part of the cohomology of a 576-dimensional Nichols algebra $\mathfrak{B}_4$ that is invariant under a natural $\mathbf{S}_4$-action. Because our counts are precise, we obtain two answers to Venkatesh's question about the topological origin of secondary terms in arithmetic counts., 57 pages, comments welcome!

Let $\mathscr {C}$ denote a finite group scheme-theoretical category over an algebraically closed field of characteristic $p\ge 0$ as introduced by the first author. For any indecomposable exact module category over $\mathscr {C}$, we classify its simple objects and provide an expression for their projective covers, in terms of double cosets and projective representations of certain closed subgroup schemes, which upgrades a result by Ostrik for group-theoretical fusion categories. As a byproduct, we describe the simples and indecomposable projectives of $\mathscr {C}$, and parametrize the Brauer-Piccard group of ${\rm Coh}(G)$ for any finite connected group scheme $G$. Finally, we apply our results to describe the blocks of the center of ${\rm Coh}(G)$, which is a group scheme-theoretical category., Comments appreciated

Let $U_q(\mathfrak{g})$ denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra $\mathfrak{g}$. Let $\lambda$ be a nonzero dominant integral weight of $\mathfrak{g}$, and let $V$ be the corresponding type $1$ finite-dimensional irreducible representation of $U_q(\mathfrak{g})$. Starting from this data, the $R$-matrix formalism for quantum groups outputs a Hopf algebra $\mathbf{U}_\mathrm{R}^\lambda(\mathfrak{g})$ defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley-Serre type presentation which can be recovered from that of $U_q(\mathfrak{g})$ by adding a single invertible quantum Cartan element. We simultaneously establish that $\mathbf{U}_\mathrm{R}^\lambda(\mathfrak{g})$ can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of $V$. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining $\mathbf{U}_\mathrm{R}^\lambda(\mathfrak{g})$, with respect to a natural grading by the root lattice of $\mathfrak{g}$ compatible with the weight space decomposition of $\mathrm{End}(V)$., Comment: 34 pages

We construct a family of unoriented 2-dimensional cobordism theories parametrized by certain triples of sequences. We also prove that some specializations of these sequences yield equivalences with an exterior product of Deligne categories. In particular, one of these specializations recovers the generalized Deligne category $\operatorname{Rep}(S_t\wr \mathbb Z_2)$., Comments welcome

We propose an algebraic viewpoint of the problem of deformation quantization of the so called almost Poisson algebras, which are algebras with a commutative associative product and an antisymmetric bracket which is a bi-derivation but does not necessarily satisfy the Jacobi identity. From that viewpoint, the main result of the paper asserts that, by contrast with Poisson algebras, the only reasonable category of algebras in which almost Poisson algebras can be quantized is isomorphic to the category of almost Poisson algebras itself, and the trivial two-term quantization formula already gives a solution to the quantization problem., 11 pages, comments are welcome

For a positive integer r, an r-spin topological quantum field theory is a 2-dimensional TQFT with tangential structure given by the r-fold cover of SO_2 . In particular, such a TQFT assigns a scalar invariant to every closed r-spin surface Sigma. Given a sequence of scalars indexed by the set of diffeomorphism classes of all such Sigma, we construct a symmetric monoidal category C and a C-valued r-spin TQFT which reproduces the given sequence. We also determine when such a sequence arises from a TQFT valued in an abelian category with finite-dimensional Hom spaces. In particular, we construct TQFTs with values in super vector spaces that can distinguish all diffeomorphism classes of r-spin surfaces, and we show that the associated algebras are necessarily non-semisimple.

We introduce group-theoretical fusion 2-categories, a strong categorification of the notion of a group-theoretical fusion 1-category. Physically speaking, such fusion 2-categories arise by gauging subgroups of a global symmetry. We show that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Then, we describe the underlying finite semisimple 2-category of group-theoretical fusion 2-categories, and, more generally, of certain 2-categories of bimodules. We also partially describe the fusion rules of group-theoretical fusion 2-categories, and investigate the group gradings of such fusion 2-categories. Using our previous results, we classify fusion 2-categories admitting a fiber 2-functor. Next, we study fusion 2-categories with a Tambara-Yamagami defect, that is $\mathbb{Z}/2$-graded fusion 2-categories whose non-trivially graded factor is $\mathbf{2Vect}$. We classify these fusion 2-categories, and examine more closely the more restrictive notion of Tambara-Yamagami fusion 2-categories. Throughout, we give many examples to illustrate our various results.

After providing a short review on the recently introduced notion of post-group by Bai, Guo, Sheng and Tang, we exhibit post-group counterparts of important post-Lie algebras in the literature, including the infinite-dimensional post-Lie algebra of Lie group integrators. The notion of free post-group is examined, and a group isomorphism between the two group structures associated to a free post-group is explicitly constructed.

In previous paper, we studied deformations $A_{\alpha,\beta}$ of the second Weyl algebra and computed their derivations. In the present paper, we identify the semiclassical limits $\mathcal{A}_{\alpha,\beta}$ of these deformations and compute their Poisson derivations. Our results show that the first Hochschild cohomology group HH$^1(A_{\alpha,\beta})$ is isomorphic to the first Poisson cohomology group HP$^1( \mathcal{A}_{\alpha,\beta}).$, Comment: 37 pages. arXiv admin note: text overlap with arXiv:2204.12214

The goal of this paper is to generalize a statement of Drinfeld that Yangians can be constructed as limit forms of the quantum loop algebras to super case. We establish the connections between quantum loop superalgebra and super Yangian of general linear Lie superalgebra $\mathfrak{gl}_{M|N}$ in $RTT$-presentation. We also apply the same argument to twisted super Yangian of the ortho-symplectic Lie superalgebra. For this purpose, we introduce the twisted quantum loop superalgebra as a one-sided coideal of quantum group superalgebra $\mathrm{U}_q(\mathfrak{gl}_{M|2n})$ with respect to the comultiplication. There is a PBW type basis of this coideal sub-superalgebra derived from its $\mathbb{A}$-form., 27 pages, 2 figures.Comments are welcome!

In this paper, we endow the family of closed oriented genus $g$ surfaces, starting with torus, with a structure of a (co)cyclic object in the category of $3$-dimensional cobordisms. As a corollary, any $3$-dimensional TQFT induces a (co)cyclic module, which we compute algebraically for the Reshetikhin-Turaev TQFT., 48 pages, many figures. Comments welcome!

We show that every involutive Hopf monoid in a complete and finitely cocomplete symmetric monoidal category gives rise to invariants of oriented surfaces defined in terms of ribbon graphs. For every ribbon graph this yields an object in the category, defined up to isomorphism, that depends only on the homeomorphism class of the associated surface. This object is constructed via (co)equalisers and images and equipped with a mapping class group action. It can be viewed as a categorical generalisation of the ground state of Kitaev's quantum double model or of a representation variety for a surface. We apply the construction to group objects in cartesian monoidal categories, in particular to simplicial groups as group objects in SSet and to crossed modules as group objects in Cat. The former yields a simplicial set consisting of representation varieties, the latter a groupoid whose sets of objects and morphisms are obtained from representation varieties., 46 pages

We define and study the shuffle algebra $Sh_{m|n}$ of the quantum toroidal algebra $\mathcal E_{m|n}$ associated to Lie superalgebra $\mathfrak{gl}_{m|n}$. We show that $Sh_{m|n}$ contains a family of commutative subalgebras $\mathcal B_{m|n}(s)$ depending on parameters $s=(s_1,\dots,s_{m+n})$, $\prod_i s_i=1$, given by appropriate regularity conditions. We show that $\mathcal B_{m|n}(s)$ is a free polynomial algebra and give explicit generators which conjecturally correspond to the traces of the $s$-weighted $R$-matrix computed on the degree zero part of $\mathcal E_{m|n}$ modules of levels $\pm 1$., Latex 31 page

We give new proofs that Khovanov homology detects the figure eight knot and the cinquefoils, and that HOMFLY homology detects $5_2$ and each of the $P(-3,3,2n+1)$ pretzel knots. For all but the figure eight these mostly follow the same lines as in previous work. The key difference is that in honor of Tom Mrowka's 60th birthday, the arguments here use instanton Floer homology rather than knot Floer homology., 17 pages; written for the proceedings of "Frontiers in Geometry and Topology," a conference in honor of Tom Mrowka's 60th birthday. v2: accepted version

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra $\mathcal A$. The algebra $\mathcal A$ is associative, noncommutative, and infinite-dimensional. It is defined by two generators $A,B$ and two relations called the down-up relations. In the present paper, we introduce the $\mathbb Z_3$-symmetric down-up algebra $\mathbb A$. We define $\mathbb A$ by generators and relations. There are three generators $A,B,C$ and any two of these satisfy the down-up relations. We describe how $\mathbb A$ is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, the $\mathfrak{sl}_3$ loop algebra, the Kac-Moody Lie algebra $A^{(1)}_2$, the $q$-Weyl algebra, the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$, and the quantized enveloping algebra $U_q (A^{(1)}_2)$. We give some open problems and conjectures., 31 pages. In Memory of Georgia Benkart (1947--2022)

Using skein theory very much in the spirit of the Reshetikhin--Turaev constructions, we define a $(3+1)$-TQFT associated with possibly non-semisimple finite unimodular ribbon tensor categories. State spaces are given by admissible skein modules, and we prescribe the TQFT on handle attachments. We give some explicit algebraic conditions on the category to define this TQFT, namely to be "chromatic non-degenerate". As a by-product, we obtain an invariant of 4-manifolds equipped with a ribbon graph in their boundary, and in the "twist non-degenerate" case, an invariant of 3-manifolds. Our construction generalizes the Crane--Yetter--Kauffman TQFTs in the semi-simple case, and the Lyubashenko (hence also Hennings and WRT) invariants of 3-manifolds. The whole construction is very elementary, and we can easily characterize invertibility of the TQFTs, study their behavior under connected sum and provide some examples., 30 pages

Over fields of characteristic zero, we determine all absolutely irreducible Yetter-Drinfeld modules over groups that have prime dimension and yield a finite-dimensional Nichols algebra. To achieve our goal, we introduce orders of braided vector spaces and study their degenerations and specializations., 22 pages

Let $\mathcal{A}$ be an associative algebra containing either classical or quantum universal enveloping algebra of a semi-simple complex Lie algebra $\mathfrak{g}$. We present a construction of the Mickelsson algebra $Z(\mathcal{A},\mathfrak{g})$ relative to the left ideal in $\mathcal{A}$ generated by positive root vectors. Our method employs a calculus on Hasse diagrams associated with classical or quantum $\mathfrak{g}$-modules. We give an explicit expression for a PBW basis in $Z(\mathcal{A},\mathfrak{g})$ in the case when $\mathcal{A}=U(\mathfrak{a})$ of a metric Lie algebra $\mathfrak{a}\supset \mathfrak{g}$. For $\mathcal{A}=U_q(\mathfrak{a})$ and $\mathfrak{g}$ the commutant of a Levi subalgebra in $\mathfrak{a}$, we construct a PBW basis in terms of quantum Lax operators, upon extension of the ground ring of scalars to $\mathbb{C}[[\hbar]]$., 19 pages, no figures

For a simple linear algebraic group $G$, the chiral universal centralizer $\mathbf{I}_{G,k}$ is a vertex operator algebra, which is the chiralization of the universal centralizer $\mathfrak{Z}_G$. The variety $\mathfrak{Z}_G$ is identified with the spectrum of the equivariant Borel-Moore homology of the affine Grassmannian of the Langlands dual group of $G$. Beem and Nair conjectured that an open symplectic immersion from $\mathrm{KT}_G$, the Kostant-Toda lattice associated to a simple group $G$, to $\mathfrak{Z}_G$ gives rises to a free field realization of the chiral universal centralizer at the critical level. In this paper, we construct a free field realization of $\mathbf{I}_{G,k}$ at any level, which coincides with the one conjectured by Beem and Nair at the critical level. We give an explicit description of this construction in $SL_2(\mathbb{C})$-case., 15 pages, revised

We propose an infinitesimal counterpart to the notion of braided category. The corresponding infinitesimal braidings are natural transformations which are compatible with an underlying braided monoidal structure in the sense that they constitute a first-order deformation of the braiding. This extends previously considered infinitesimal symmetric or Cartier categories, where involutivity of the braiding and an additional commutativity of the infinitesimal braiding with the symmetry are required. The generalized pre-Cartier framework is then elaborated in detail for the categories of (co)quasitriangular bialgebra (co)modules and we characterize the resulting infinitesimal $\mathcal{R}$-matrices (resp. $\mathcal{R}$-forms) on the bialgebra. It is proven that the latter are Hochschild $2$-cocycles and that they satisfy an infinitesimal quantum Yang-Baxter equation, while they are Hochschild $2$-coboundaries under the Cartier (co)triangular assumption in the presence of an antipode. We provide explicit examples of infinitesimal braidings, particularly on quantum $2\times 2$-matrices, $\mathrm{GL}_q(2)$, Sweedler's Hopf algebra and via Drinfel'd twist deformation. As conceptual tools to produce examples of infinitesimal braidings we prove an infinitesimal version of the FRT construction and we provide a Tannaka-Krein reconstruction theorem for pre-Cartier coquasitriangular bialgebras. We comment on the deformation of infinitesimal braidings and construct a quasitriangular structure on formal power series of Sweedler's Hopf algebra., 38 pages

We study the quantum vertex algebraic framework for the Yangians of RTT-type and the braided Yangians associated with Hecke symmetries, introduced by Gurevich and Saponov. First, we construct several families of modules for the aforementioned Yangian-like algebras which, in the RTT-type case, lead to a certain $h$-adic quantum vertex algebra $\mathcal{V}_c (R)$ via the Etingof-Kazhdan construction, while, in the braided case, they produce ($\phi$-coordinated) $\mathcal{V}_c (R)$-modules. Next, we show that the coefficients of suitably defined quantum determinant can be used to obtain central elements of $\mathcal{V}_c (R)$, as well as the invariants of such ($\phi$-coordinated) $\mathcal{V}_c (R)$-modules. Finally, we investigate a certain algebra which is closely connected with the representation theory of $\mathcal{V}_c (R)$., Comment: 23 pages, comments are welcome

In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(\mathfrak g,r)$ on a smooth manifold $M$. Using the formality of polydifferential operators on Lie algebroids we obtain a deformation quantization of $M$ together with a quantum group $\mathscr{U}_\hbar(\mathfrak{g})$ and a map of associated DGLA's. This motivates a definition of quantum action in terms of $L_\infty$-morphisms which generalizes the one given by Drinfeld., 26 pages. Comments are welcome!

We study the Gram determinant and construct bases of hom spaces for the one-dimensional topological theory of decorated unoriented one-dimensional cobordisms, as recently defined by Khovanov, when the pair of generating functions is linear., 14 pages

In this paper, we review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras, which have attracted quite much interest recently. Using Voronov's higher derived brackets, one can obtain an $L_\infty$-algebra whose Maurer-Cartan elements are relative Rota-Baxter Lie algebras. Then using the twisting method, one can obtain the $L_\infty$-algebra that controls deformations of a relative \RB Lie algebra. Meanwhile, the cohomologies of relative Rota-Baxter Lie algebras can also be defined with the help of the twisted $L_\infty$-algebra. Using the controlling algebra approach, one can also introduce the notion of homotopy relative Rota-Baxter Lie algebras with close connection to pre-Lie$_\infty$-algebras. Finally, we briefly review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras of nonzero weights., Comment: arXiv admin note: substantial text overlap with arXiv:2008.06714

We generalize the notion of spectral triple with reality structure to spectral triples with multitwisted real structure, the class of which is closed under the tensor product composition. In particular, we introduce a multitwisted order one condition (characterizing the Dirac operators as an analogue of first-order differential operator). This provides a unified description of the known examples, which include rescaled triples with the conformal factor from the commutant of the algebra and (on the algebraic level) triples on quantum disc and on quantum cone, that satisfy twisted first order condition of \cite{BCDS16,BDS19}, as well as asymmetric tori, non-scalar conformal rescaling and noncommutative circle bundles. In order to deal with them we allow twists that do not implement automorphisms of the algebra of spectral triple., 12 pages

In continuation of our recent work arXiv:2006.07312, we classify the extremal traces on infinite diagram algebras that appear in the context of Schur-Weyl duality for Banica and Speicher's easy groups. We show that the branching graphs of these algebras describe walks on new variations of the Young graph which describe curious ways of growing Young diagrams. As a consequence, we prove that the extremal traces on generic rook-Brauer algebras are always extensions of extremal traces on the group algebra $\mathbb{C}[S_{\infty}]$ of the infinite symmetric group. Moreover, we conjecture that the same is true for generic parameter deformations of the centralizers of the hyperoctahedral group and we reduce this conjecture to a conceptually much simpler numerical statement. Lastly, we address the trace classification problem for the Schur-Weyl dual of the halfliberated orthogonal group $O_N^*$, in which case extremal traces are always extensions of extremal traces on $\mathbb{C}[S_{\infty} \times S_{\infty}]$. Our approach relies on methods developed by Vershik and Nikitin., Comment: Comments welcome!

Let $\mathcal{A}$ and $\mathcal{B}$ be monoidal categories and let $R:\mathcal{A} \rightarrow \mathcal{B}$ be a lax monoidal functor. If $R$ has a left adjoint $L$, it is well-known that the two adjoints induce functors $\overline{R}={\sf Alg}(R):{\sf Alg}(\mathcal{A})\rightarrow {\sf Alg }(\mathcal{B})$ and $\underline{L}={\sf Coalg(L)}:{\sf Coalg}(\mathcal{B})\rightarrow {\sf Coalg}(\mathcal{A})$ respectively. The pair $(L,R)$ is called "liftable" if the functor $\overline{R}$ has a left adjoint and if the functor $\underline{L}$ has a right adjoint. A pleasing fact is that, when $\mathcal{A}$, $\mathcal{B}$ and $R$ are moreover braided, a liftable pair of functors as above gives rise to an adjunction at the level of bialgebras. In this note, sufficient conditions on the category $\mathcal{A}$ for $\overline{R}$ to possess a left adjoint, are given. Natively these conditions involve the existence of suitable colimits that we interpret as objects which are simultaneously initial in four distinguished categories (among which the category of epi-induced objects), allowing for an explicit construction of $\overline{L}$, under the appropriate hypotheses. This is achieved by introducing a relative version of the notion of weakly coreflective subcategory, which turns out to be a useful tool to compare the initial objects in the involved categories. We apply our results to obtain an analogue of Sweedler's finite dual for the category of vector spaces graded by an abelian group $G$ endowed with a bicharacter. When the bicharacter on $G$ is skew-symmetric, a lifted adjunction as mentioned above is explicitly described, inducing an auto-adjunction on the category of bialgebras "colored" by $G$., The previous verision has been revised by means of weak coreflections and initial objects. The study of pre-rigid categories has been extrapolated and expanded to become an independent research line (arXiv:2201.03952)