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The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. We investigate the alternating central extension $\mathcal O_q$ of $O_q$. The algebra $\mathcal O_q$ was introduced by Baseilhac and Koizumi, who called it the current algebra of $O_q$. Recently Baseilhac and Shigechi gave a presentation of $\mathcal O_q$ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of $\mathcal O_q$ that involves a subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of $\mathcal O_q$. This presentation resembles the compact presentation of the alternating central extension for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$., Comment: 26 pages. arXiv admin note: text overlap with arXiv:2103.03028

We establish the Schur-Weyl type duality between double affine Hecke algebras and quantum toroidal superalgebras, generalizing the well known result of Vasserot-Varagnolo [VV96] to the super case., Comment: 17 pages; v3 a new Proposition 2.7 to fix a gap in the proof of Proposition 2.9; to appear in Journal of Pure and Applied Algebra

L. N. Childs defined a bi-skew brace to be a skew brace such that if we swap the role of the two operations, then we find again a skew brace. In this paper, we give a systematic analysis of bi-skew braces. We study nilpotency and solubility, and connections between bi-skew braces and set-theoretic solutions of the Yang--Baxter equation. Further, we deal with Byott's conjecture in the case of bi-skew braces, and we use bi-skew braces as a tool to solve a classification problem proposed by L. Vendramin. In the final part, we investigate brace blocks, defined by A. Koch to be families of group operations on a given set such that any two of them yield a bi-skew brace. We provide a characterisation of brace blocks, illustrate how all known constructions in literature follow in a natural way from our characterisation, and give several new examples., Comment: Final version, published in Journal of Pure and Applied Algebra

We introduce the notions of sovereign, spherical and balanced quasi-Hopf algebra. We investigate the connections between these, as well as their connections with the class of pivotal, involutory and ribbon quasi-Hopf algebras, respectively. Examples of balanced and ribbon quasi-Hopf algebras are obtained from a sort of double construction which associates to a braided category (resp. rigid braided) a balanced (resp. ribbon) one., 25 pages with figures, submitted

We use the machinery of categorified Jones-Wenzl projectors to construct a categorification of a type A Reshetikhin-Turaev invariant of oriented framed tangles where each strand is labeled by an arbitrary finite-dimensional representation. As a special case, we obtain a categorification of the coloured Jones polynomial of links.

Degenerating the quantum queer Schur superalgebra ${\mathcal{Q}_q(n,r; R)}$ to the case $q=1$, the queer Schur superalgebra ${\mathcal{Q}(n,r)}$ is obtained. In this article, we reconstruct the universal enveloping algebra ${U({\mathfrak{q}_n})}$ of the queer Lie superalgebra ${\mathfrak{q}_n}$ via ${\mathcal{Q}(n,r)}$, and achieve another explanation of the Schur-Weyl-Sergeev duality. Finally, we depict the Schur-Weyl-Sergeev duality over $\mathbb{Z}$.

We establish a multiplication formula for a tridiagonal standard basis element in the idempotented coideal subalgebras of quantum affine $\mathfrak{gl}_n$ arising from the geometry of affine partial flag varieties of type $C$. We apply this formula to obtain the stabilization algebras $\dot{\mathbf K}^{\mathfrak{c}}_n$, $\dot{\mathbf K}^{\jmath \imath}_{\mathfrak{n}}$, $\dot{\mathbf K}^{\imath \jmath}_{\mathfrak{n}}$ and $\dot{\mathbf K}^{\imath \imath}_{\eta}$, which are idempotented coideal subalgebras of quantum affine $\mathfrak{gl}_n$. The symmetry in the formula leads to an isomorphism of the idempotented coideal subalgebras $\dot{\mathbf K}^{\jmath \imath}_{\mathfrak{n}}$ and $\dot{\mathbf K}^{\imath \jmath}_{\mathfrak{n}}$ with compatible monomial, standard and canonical bases., Comment: 39 pages, fixed some typos, to appear in Journal of Pure and Applied Algebra

The Frobenius-Perron dimension for an abelian category was recently introduced. We apply this theory to the category of representations of the finite-dimensional radical squared zero algebras associated to certain modified ADE graphs. In particular, we take an ADE quiver with arrows in a certain orientation and an arbitrary number of loops at each vertex. We show that the Frobenius-Perron dimension of this category is equal to the maximum number of loops at a vertex. Along the way, we introduce a result which can be applied in general to calculate the Frobenius-Perron dimension of a radical square zero bound quiver algebra. We use this result to introduce a family of abelian categories which produce arbitrarily large irrational Frobenius-Perron dimensions., Comment: typos corrected

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G \to H$ of finite groups assigns in a functorial way to a $G$-equivariant topological field theory an $H$-equivariant topological field theory, the pushforward theory. When $H$ is the trivial group, this yields an orbifold construction for $G$-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization., 21 pages, accepted for publication in the Journal of Pure and Applied Algebra

We study unitary pseudonatural transformations (UPTs) between fibre functors Rep(G) -> Hilb, where G is a compact quantum group. For fibre functors F_1, F_2 we show that the category of UPTs F_1 -> F_2 and modifications is isomorphic to the category of finite-dimensional *-representations of the corresponding bi-Hopf-Galois object. We give a constructive classification of fibre functors accessible by a UPT from the canonical fibre functor, as well as UPTs themselves, in terms of Frobenius algebras in the category Rep(A_G), where A_G is the Hopf *-algebra dual to the compact quantum group. As an example, we show that finite-dimensional quantum isomorphisms from a quantum graph X are UPTs between fibre functors on Rep(G_X), where G_X is the quantum automorphism group of X., 53 pages, many coloured pictures | Rev 7: Final version

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra T p for any integer p > 2 which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of T p , as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology.

The aim of this note is to understand the injectivity of Feigin's map $\mathbf{F_w}$ by representation theory of quivers, where $\mathbf{w}$ is the word of a reduced expression of the longest element of a finite Weyl group. This is achieved by the Ringel-Hall algebra approach and a careful studying of a well-knwon total order on the category of finite-dimensional representations of a valued quiver of finite type. As a byproduct, we also generalize Reineke's construction of monomial bases to non-simply-laced cases., Comment: to appear in JPAA

We prove there exist infinitely many inequivalent fusion categories whose Grothendieck rings do not admit any pseudounitary categorifications., Comment: 17 pages

In this paper, we classify all simple jet modules for the Neveu-Schwarz algebra $\widehat{\mathfrak{k}}$ and its contact subalgebra $\mathfrak{k}^+$. Based on these results, we give a classification of simple Harish-Chandra modules for $\widehat{\mathfrak{k}}$ and $\mathfrak{k}^+$., Comment: arXiv admin note: text overlap with arXiv:2007.04483

We define a symmetric tensor product on the Drinfeld centre of a symmetric fusion category, in addition to its usual tensor product. We examine what this tensor product looks like under Tannaka duality, identifying the symmetric fusion category with the representation category of a finite (super)-group. Under this identification, the Drinfeld centre is the category of equivariant vector bundles over the finite group (underlying the super-group, in the super case). In the non-super case, we show that the symmetric tensor product corresponds to the fibrewise tensor product of these vector bundles. In the super case, we define for each super-group structure on the finite group a super-version of the fibrewise tensor product. We show that the symmetric tensor product on the Drinfeld centre of the representation category of the resulting finite super-groups corresponds to this super-version of the fibrewise tensor product on the category of equivariant vector bundles over the finite group., Comment: 40 pages, many diagrams

We associate a diagrammatic monoidal category $\mathcal{H}\textit{eis}_k(A;z,t)$, which we call the quantum Frobenius Heisenberg category, to a symmetric Frobenius superalgebra $A$, a central charge $k \in \mathbb{Z}$, and invertible parameters $z,t$ in some ground ring. When $A$ is trivial, i.e. it equals the ground ring, these categories recover the quantum Heisenberg categories introduced in our previous work, and when the central charge $k$ is zero they yield generalizations of the affine HOMFLY-PT skein category. By exploiting some natural categorical actions of $\mathcal{H}\textit{eis}_k(A;z,t)$ on generalized cyclotomic quotients, we prove a basis theorem for morphism spaces., Comment: 44 pages

We show how to construct all the extensions of left braces by ideals with trivial structure. This is useful to find new examples of left braces. But, to do so, we must know the basic blocks for extensions: the left braces with no ideals except the trivial and the total ideal, called simple left braces. In this article, we present the first non-trivial examples of finite simple left braces. To explicitly construct them, we define the matched product of two left braces, which is a useful method to recover a finite left brace from its Sylow subgroups., Comment: References in the previous version were broken

Let $p$ and $q$ be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $pq^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac-Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix., Remark 6.9 added; notation change in equation (5.7); some typos corrected and references updated. Accepted in J. Pure Appl. Algebra

Circuit algebras, introduced by Bar-Natan and the first author, are a generalization of Jones's planar algebras, in which one drops the planarity condition on "connection diagrams". They provide a useful language for the study of virtual and welded tangles in low-dimensional topology. In this note, we present the circuit algebra analogue of the well-known classification of planar algebras as pivotal categories with a self-dual generator. Our main theorem is that there is an equivalence of categories between circuit algebras and the category of linear wheeled props - a type of strict symmetric tensor category with duals that arises in homotopy theory, deformation theory and the Batalin-Vilkovisky quantization formalism., Comment: 29 pages, many figures

We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form $A = \kk Q/I$, where $Q$ is a quiver and $I$ is an ideal of relations coming from taking partial derivatives of a twisted superpotential on $Q$. We define the type $(M, P, d)$ of such an algebra $A$, where $M$ is the incidence matrix of the quiver, $P$ is the permutation matrix giving the action of the Nakayama automorphism of $A$ on the vertices of the quiver, and $d$ is the degree of the superpotential. We study the question of what possible types can occur under the additional assumption that $A$ has polynomial growth. In particular, we are able to give a nearly complete answer to this question when $Q$ has at most 3 vertices., Comment: Small changes throughout

We provide a general construction of integral TQFTs over a general commutative ring, $\mathbf{k}$, starting from a finite Hopf algebra over $\mathbf{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds. We show the construction applies to index 2 extensions of the Borel parts of Lusztig's small quantum groups for all simple Lie types, yielding integral TQFTs over the cyclotoic integers for surfaces with boundary. We further establish and compute isomorphisms of TQFT functors constructed from Hopf algebras that are related by a strict gauge transformation in the sense of Drinfeld. Formulas for the natural isomorphisms are given in terms of the gauge twist element. These results are combined and applied to show that the Hennings invariant associated to quantum-$sl_2$ takes values in the cyclotomic integers. Using prior results of Chen et al we infer integrality also of the Witten-Reshetikhin-Turaev $SO(3)$ invariant for rational homology spheres. As opposed to most other approaches the methods described in this article do not invoke calculations of skeins, knots polynomials, or representation theory, but follow a combinatorial construction that uses only the elements and operations of the underlying Hopf algebras., Algebraic criteria for underlying Hopf algebra and commutative ring have been corrected, including several lemmas, propositions, and theorems depending on these. (The prior Dedekind condition did not suffice due to possible non-trivial Picard groups. It has been replaced by a Frobenius condition)

The irreducible spin character values of the wreath products of the hyperoctahedral groups with an arbitrary finite group are determined., Comment: 19 pages, 1 table

We show that in ADE type the trace of Webster's categorification of a tensor product of irreducibles for the quantum group is isomorphic to a tensor product of Weyl modules for the current algebra $\dot{U}(\mathfrak{g}[t])$. This extends a result of Beliakova, Habiro, Lauda, and Webster who showed that the trace of the categorified quantum group $\dot{\mathcal{U}}^*(\mathfrak{g})$ is isomorphic to $\dot{U}(\mathfrak{g}[t])$, and the trace of a cyclotomic quotient of $\dot{\mathcal{U}}^*(\mathfrak{g})$, which categorifies a single irreducible for the quantum group, is isomorphic to a Weyl module for $\dot{U}(\mathfrak{g}[t])$. We use a deformation argument based on Webster's technique of unfurling 2-representations., Comment: 41 pages; preliminary version, comments welcome

A symmetric bilinear form on a certain subspace $\widehat{\mathbb T}^{\bf b}$ of a completion of the Fock space $\mathbb T^{{\bf b}}$ is defined. The canonical and dual canonical bases of $\widehat{\mathbb T}^{\bf b}$ are dual with respect to the bilinear form. As a consequence, the inversion formula connecting the coefficients of the canonical basis and that of the dual canonical basis of $\widehat{\mathbb T}^{\bf b}$ expanded in terms of the standard monomial basis of $\mathbb T^{{\bf b}}$ is obtained. Combining with the Brundan's algorithm for computing the elements in the canonical basis of $\widehat{\mathbb{T}}^{{\bf b}_{\mathrm{st}}}$, we have an algorithm computing the elements in the canonical basis of $\widehat{\mathbb{T}}^{{\bf b}}$ for arbitrary ${\bf b}$., Comment: 23 pages

To a graded finite-rank matrix factorisation of the difference of two homogeneous potentials one can assign two numbers, the left and right quantum dimension. The existence of such a matrix factorisation with non-zero quantum dimensions defines an equivalence relation between potentials, giving rise to non-obvious equivalences of categories. Restricted to ADE singularities, the resulting equivalence classes of potentials are those of type {A_{d-1}} for d odd, {A_{d-1},D_{d/2+1}} for d even but not in {12,18,30}, and {A_{11}, D_7, E_6}, {A_{17}, D_{10}, E_7} and {A_{29}, D_{16}, E_8}. This is the result expected from two-dimensional rational conformal field theory, and it directly leads to new descriptions of and relations between the associated (derived) categories of matrix factorisations and Dynkin quiver representations., 29 pages

In this paper we study categories of tilting modules. Our starting point is the tilting modules for a reductive algebraic group G in positive characteristic. Here we extend the main result in [8] by proving that these tilting modules form a (strictly object-adapted) cellular category. We use this result to specify a subset of cellular basis elements, which generates all morphisms in this category. In a different direction we generalize the earlier results to the case where G is replaced by the infinitesimal thickenings G_rT of a maximal torus T in G by the Frobenius subgroup schemes G_r. Here our procedure leads to a special set of generators for the morphisms in the category of projective G_rT- modules. Our methods are rather general (applying to "quasi hereditary like" categories). In particular, there are completely analogous results for tilting modules of quantum groups at roots of unity. As examples we treat the tilting modules in the ordinary BGG category O, and in the modular case we examine G = SL_2 in some details., 32 pages, with corrections and more details

Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we describe those Lie superalgebras which have a faithful finite dimensional unitary representation. We call these Lie superalgebras unitary. This is achieved by describing the classification of real finite dimensional compact simple Lie superalgebras, and analyzing, in a rather elementary and direct way, the decomposition of reductive Lie superalgebras ($\g$ is a semisimple $\g_{\bar 0}$-module) over fields of characteristic zero into ideals., 22 pages

In this paper, the Drinfeld center of a monoidal category is generalized to a class of mixed Drinfeld centers. This gives a unified picture for the Drinfeld center and a natural Heisenberg analogue. Further, there is an action of the former on the latter. This picture is translated to a description in terms of Yetter-Drinfeld and Hopf modules over quasi-bialgebras in a braided monoidal category. Via braided reconstruction theory, intrinsic definitions of braided Drinfeld and Heisenberg doubles are obtained, together with a generalization of the result in [Lu94] that the Heisenberg double is a 2-cocycle twist of the Drinfeld double for general braided Hopf algebras., Comment: 76 pages, minor corrections to V2

Let M be a graded Lie algebra, together with graded Lie subalgebras L and A such that as a graded space M is the direct sum of L and A, and A is abelian. Let D be a degree one derivation of M squaring to zero and sending L into itself, then Voronov's construction of higher derived brackets associates to D a L-infinity structure on A[-1]. It is known, and it follows from the results of this paper, that the resulting L-infinity algebra is a weak model for the homotopy fiber of the inclusion of differential graded Lie algebras i : (L,D,[, ]) -> (M,D,[, ]). We prove this fact using homotopical transfer of L-infinity structures, in this way we also extend Voronov's construction when the assumption A abelian is dropped: the resulting formulas involve Bernoulli numbers. In the last section we consider some example and some further application., v3: several changes in the exposition

In this paper we study fields satisfying $N$-point locality and their properties. We obtain residue formulae for $N$-point local fields in terms of derivatives of delta functions and Bell polynomials. We introduce the notion of the space of descendants of $N$-point local fields which includes normal ordered products and coefficients of operator product expansions. We show that examples of $N$-point local fields include the vertex operators generating the boson-fermion correspondences of type B, C and D-A. We apply the normal ordered products of these vertex operators to the setting of the representation theory of the double-infinite rank Lie algebras $b_{\infty}, c_{\infty}, d_{\infty}$. Finally, we show that the field theory generated by $N$-point local fields and their descendants has a structure of a twisted vertex algebra., long version with additional details and proofs

For any finite-dimensional Hopf algebra $H$ we construct a group homomorphism $\biga(H)\to \text{BrPic}(\Rep(H))$, from the group of equivalence classes of $H$-biGalois objects to the group of equivalence classes of invertible exact $\Rep(H)$-bimodule categories. We discuss the injectivity of this map. We exemplify in the case $H=T_q$ is a Taft Hopf algebra and for this we classify all exact indecomposable $\Rep(T_q)$-bimodule categories., 26 pages. Accepted in Journal of Pure and Applied Algebra

Under the assumption that the quantum parameter $q$ is an $l$-th primitive root of unity with $l$ odd in a field $F$ of characteristic 0 and $m+n\geq r$, we obtained a complete classification of irreducible modules of the $q$-Schur superalgebra introduced H. Rui and the first Author., 50 pages

We continue studying properties of semisimple Hopf algebras $H$ over algebraically closed fields of characteristic 0 resulting from their generalized character tables. We show that the generalized character table of $H$ reflect normal left coideal subalgebras of $H.$ These are the Hopf analogues of normal subgroups in the sense that they arise from Hopf quotients. We apply these ideas to prove Hopf analogues of known results in group theory. Among the rest we prove that columns of the character table are orthogonal and that all entries are algebraic integers. We analyze `semi kernels' and their relations to the character table. We prove a full analogue of the Burnside-Brauer theorem for almost cocommutative $H.$ We also prove the Hopf algebras analogue of the following (Burnside) theorem: If G is a non-abelian simple group then $\{1\}$ is the only conjugacy class of G which has prime power order., 27 pages

A description of all normal Hopf subalgebras of a semisimple Drinfeld double is given. This is obtained by considering an analogue of Goursat's lemma concerning fusion subcategories of Deligne products of two fusion categories. As an application we show that the Drinfeld double of any abelian extension is also an abelian extension., Title changed, 23 pages. Comments are welcomed!

In this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra without vacuum which is universal to the forgetful functor. Furthermore, from any Leibniz algebra $\g$ we construct a vertex Leibniz algebra $V_{\g}$ and show that $V_{\g}$ can be embedded into a vertex algebra if and only if $\g$ is a Lie algebra., Comment: latex, 25 pages

We give a classification of Wada-type representations of the braid groups, and solutions of a variant of the set-theoretical Yang-Baxter equation adapted to the free-product group structure. As a consequence, we prove Wada's conjecture: There are only seven types of Wada-type representations up to certain symmetries., Comment: 13 pages, 2 figures: Added Lemma 2.2, which was implicit in the previous version without proof and more explanations

Consider a pronilpotent DG (differential graded) Lie algebra over a field of characteristic 0. In the first part of the paper we introduce the reduced Deligne groupoid associated to this DG Lie algebra. We prove that a DG Lie quasi-isomorphism between two such algebras induces an equivalence between the corresponding reduced Deligne groupoids. This extends the famous result of Goldman- Millson (attributed to Deligne) to the unbounded pronilpotent case. In the second part of the paper we consider the Deligne 2-groupoid. We show it exists under more relaxed assumptions than known before (the DG Lie algebra is either nilpotent or of quasi quantum type). We prove that a DG Lie quasi-isomorphism between such DG Lie algebras induces a weak equivalence between the corresponding Deligne 2-groupoids. In the third part of the paper we prove that an L-infinity quasi-isomorphism between pronilpotent DG Lie algebras induces a bijection between the sets of gauge equivalence classes of Maurer-Cartan elements. This extends a result of Kontsevich and others to the pronilpotent case., Comment: 23 pages, 2 figures. New version: change of title, other minor changes. Version 3: 31 pages, added material on the Deligne 2-groupoid. Version 4: minor changes, final, to appear in JPAA

We construct rank varieties for the Drinfel'd double of the Taft algebra and for U_q(sl2). For the Drinfel'd double when n=2 this uses a result which identifies a family of subalgebras that control projectivity of A-modules whenever A is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext^*(M,M) is finitely generated over the cohomology ring of the Drinfel'd double for any finitely-generated module M., Comment: 13 pages, submitted

Given a Lie bialgebra (g,g*), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g*. We write down families of examples for the case that g is a classical complex simple Lie algebra., Comment: 12 pages. New, shorter proof for Prop. 2.3. New Remark 4.1. Some material moved to appendix. Final version

We give a fast algorithm for computing the canonical basis of an irreducible highest-weight module for U q ( s l e ) , generalising the LLT algorithm.

In previous work, the author proved that there is a countably infinite family of N=2 superconformal equivalence classes of DeWitt N=2 superconformal super-Riemann surfaces with closed, genus-zero body. In this paper, we determine the automorphism groups for these N=2 superconformal super-Riemann surfaces, and analyze the Lie structure of these groups. Under the correspondence between N=2 superconformal and N=1 superanalytic structures, the results extend to the determination of automorphism groups of N=1 superanalytic DeWitt super-Riemann surfaces with closed, genus-zero body., Comment: Corollary 6.1 renamed a Theorem; minor adjustments. Final version; to appear in J. Pure Appl. Alg.

For a braided vector space $(V,\sigma)$ with braiding $\sigma$ of Hecke type, we introduce three associative algebra structures on the space $\oplus_{p=0}^{M}\mathrm{End}S_\sigma^p(V)$ of graded endomorphisms of the quantum symmetric algebra $S_\sigma(V)$. We use the second product to construct a new trace. This trace is an algebra morphism with respect to the third product. In particular, when $V$ is the fundamental representation of $\mathcal{U}_{q}\mathfrak{sl}_{N+1}$ and $\sigma$ is the action of the $R$-matrix, this trace is a scalar multiple of the quantum trace of type $A$., Comment: 19 pages;typos added; version for the publication of JPAA before proof

In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules.

We study twisted modules for (weak) quantum vertex algebras and we give a conceptual construction of (weak) quantum vertex algebras and their twisted modules. As an application we construct and classify irreducible twisted modules for a certain family of quantum vertex algebras., 31 pages

The Hochschild homology/cohomology an associative algebra, together with the Connes differential, the contraction map and the Lie derivative, forms the structure of calculus. In this paper we compute explicitely the calculus structure of preprojective algebras of Dynkin quivers over a field of characteristic zero. This also completes the work in math.AG/0502301, where the Batalin-Vilkovisky structure of the Hochschild cohomology of preprojective algebras of non-Dynkin quivers are computed and the calculus can be easily computed from that., Comment: 30 pages

In this paper, we prove that the cyclotomic BMW algebras B2p+1,n are cellular in the sense of [16]. We also classify the irreducible B2p+1,nmodules over a field., 48 pages. submitted, We delete one sentence at the end of paragraph 2 in page 1, and add one reference

We give a general description of the structure of a discrete double groupoid (with an extra, quite natural, filling condition) in terms of groupoid factorizations and groupoid 2-cocycles with coefficients in abelian group bundles. Our description goes as follows: To any double groupoid, we associate an abelian group bundle and a second double groupoid, its frame. The frame satisfies that every box is determined by its edges, and thus is called a `slim' double groupoid. In a first step, we prove that every double groupoid is obtained as an extension of its associated abelian group bundle by its frame. In a second, independent, step we prove that every slim double groupoid with filling condition is completely determined by a factorization of a certain canonically defined `diagonal' groupoid., Comment: amslatex, 28 pages, revised version to appear in J. Pure Appl. Algebra

We study vertex algebras and their modules associated with possibly degenerate even lattices, using an approach somewhat different from others. Several known results are recovered and a number of new results are obtained. We also study modules for Heisenberg algebras and we classify irreducible modules satisfying certain conditions and obtain a complete reducibility theorem., 24 pages

We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on vertex coalgebras. The former two properties require grading. We then discuss comodule structure. We conclude by discussing instances where graded vertex coalgebras appear, particularly as related to Primc's vertex Lie algebra and (universal) enveloping vertex algebras., 26 pages

We consider an interesting class of braidings defined by a combinatorial property in an earlier paper. We show that it consists exactly of those braidings that come from certain Yetter-Drinfeld module structures over pointed Hopf algebras with abelian coradical. As a tool we define a reduced version of the FRT construction. For braidings induced by U_q(g)-modules the reduced FRT construction is calculated explicitly., Comment: 20 pages