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We show that if X_n is a variety of cxn-matrices that is stable under the group Sym([n]) of column permutations and if forgetting the last column maps X_n into X_{n-1}, then the number of Sym([n])-orbits on irreducible components of X_n is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine FI^op-schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one FI^op-scheme becomes of product form, where X_n=Y^n for some scheme Y in affine c-space. Furthermore, to any FI^op-scheme of width one we associate a component functor from the category FI of finite sets with injections to the category PF of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that Sym([n])-orbits of components of X_n, for all n, correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem., final version, 40 pages, added several examples and clarifications and corrected typos---with many thanks to a referee!

Let $\mathsf{T}$ be a triangulated category with shift functor $\Sigma \colon \mathsf{T} \to \mathsf{T}$. Suppose $(\mathsf{A},\mathsf{B})$ is a co-t-structure with coheart $\mathsf{S} = \Sigma \mathsf{A} \cap \mathsf{B}$ and extended coheart $\mathsf{C} = \Sigma^2 \mathsf{A} \cap \mathsf{B} = \mathsf{S} * \Sigma \mathsf{S}$, which is an extriangulated category. We show that there is a bijection between co-t-structures $(\mathsf{A}',\mathsf{B}')$ in $\mathsf{T}$ such that $\mathsf{A} \subseteq \mathsf{A}' \subseteq \Sigma \mathsf{A}$ and complete cotorsion pairs in the extended coheart $\mathsf{C}$. In the case that $\mathsf{T}$ is Hom-finite, $\mathbf{k}$-linear and Krull-Schmidt, we show further that there is a bijection between complete cotorsion pairs in $\mathsf{C}$ and functorially finite torsion pairs in $\mathsf{mod}\, \mathsf{S}$., Comment: 15 pages, 2 figures

The aim of this paper is to study, by using the mathematical tools developed by Chałupnik, Touzé, and Van der Kallen, the effect of the Frobenius twist on $\operatorname{Ext}$-group in the category of strict polynomial functors. As an application, we obtain explicit formulas of cohomology of the orthogonal groups and symplectic ones.

We give a complete combinatorial answer to Kostant's problem for simple highest weight modules indexed by fully commutative permutations. We also propose a reformulation of Kostant's problem in the context of fiab bicategories and classify annihilators of simple objects in the principal birepresentations of such bicategories generalising the Barbasch--Vogan theorem for Lie algebras., Comment: Minor revision following comments by referees. To appear in Rev. Mat. Iberoam

We study the algebraic $$K\!$$ K -theory and Grothendieck–Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $$K\!$$ K -theory space of an integral monoid scheme X in terms of its Picard group $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) and pointed monoid of regular functions $$\Gamma (X, {\mathcal {O}}_X)$$ Γ ( X , O X ) and a complete description of the Grothendieck–Witt space of X in terms of an additional involution on $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) . We also prove space-level projective bundle formulae in both settings.

A pair (A, P) is called a cover of EndA(P)op if the Schur functor HomA(P,-) is fully faithful on the full subcategory of projective A-modules, for a given projective A-module P. By definition, Morita algebras are the covers of self-injective algebras and then P is a faithful projective-injective module. Conversely, we show that A is a Morita algebra and EndA(P)op is self-injective whenever (A, P) is a cover of EndA(P)op for a faithful projective-injective module P., Studienstiftung des Deutschen Volkes, Projekt DEAL

We characterize cyclic algebras over the associative and the framed little 2-disks operad in any symmetric monoidal bicategory. The cyclicity is appropriately treated in a coherent way, i.e. up to coherent isomorphism. When the symmetric monoidal bicategory is specified to be a certain symmetric monoidal bicategory of linear categories subject to finiteness conditions, we prove that cyclic associative and cyclic framed little 2-disks algebras, respectively, are equivalent to pivotal Grothendieck-Verdier categories and ribbon Grothendieck-Verdier categories, a type of category that was introduced by Boyarchenko-Drinfeld based on Barr's notion of a $\star$-autonomous category. We use these results and Costello's modular envelope construction to obtain two applications to quantum topology: I) We extract a consistent system of handlebody group representations from any ribbon Grothendieck-Verdier category inside a certain symmetric monoidal bicategory of linear categories and show that this generalizes the handlebody part of Lyubashenko's mapping class group representations. II) We establish a Grothendieck-Verdier duality for the category extracted from a modular functor by evaluation on the circle (without any assumption on semisimplicity), thereby generalizing results of Tillmann and Bakalov-Kirillov., 66 pages, lots of figures and diagrams; v3: minor changes in terminology to be consistent with follow-up papers (ribbon GV instead of balanced braided GV, auxiliary notion of self-dual algebra), comment on invertibility of maps of modular algebras included, minor changes in the presentation in several places; v4: some changes following comments by the referee, to appear in Quart. J. Math

The goal of this paper is to construct trace maps for the six functor formalism of motivic cohomology after Voevodsky, Ayoub, and Cisinski-D\'{e}glise. We also construct an $\infty$-enhancement of such a trace formalism. In the course of the $\infty$-enhancement, we need to reinterpret the trace formalism in a more functorial manner. This is done by using Suslin-Voevodsky's relative cycle groups.

We introduce a notion of globular multicategory with homomorphism types. These structures arise when organizing collections of "higher category-like" objects such as type theories with identity types. We show how these globular multicategories can be used to construct various weak higher categorical structures of types and terms., Comment: 25 pages

Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studied in Matsui-Nam-Takahashi-Tri-Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R. In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), and explore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrations of subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi and Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria., Comment: 17 pages, to appear in IMRN

Some new characterizations of the generalized Moore-Penrose inverse are proposed using range, null space, several matrix equations and projectors. Several representations of the generalized Moore-Penrose inverse are given. The relationships between the generalized Moore-Penrose inverse and other generalized inverses are discussed using core-EP decomposition. The generalized Moore-Penrose matrices are introduced and characterized. One relation between the generalized Moore-Penrose inverse and corresponding nonsingular border matrix is presented. In addition, applications of the generalized Moore-Penrose inverse in solving restricted matrix equations are studied.

We introduce a concept of connectedness of antipodal sets of compact Riemannian symmetric spaces and construct a method to make a bigger antipodal set from a given antipodal set. Moreover, using the connectedness we give a sufficient condition that a given maximal antipodal set is homogeneous.

In this note, we study the connection between Gray tensor product and suspension. We derive a characterization of weak equivalences as fully faithful and essentially surjective functors. We construct the $co$ duality, a weak involution that reverses the direction of even dimensional cells. We conclude by studying Grothendieck fibrations of complicial sets., Comment: This text is still in draft form. A new version should quickly replace this one

We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced \cite{AGV1} bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of $A$, called $\Omega$-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study $V$-universal measuring coalgebras and $V$-universal comeasuring algebras between $\Omega$-algebras $A$ and $B$, relative to a fixed subspace $V$ of $\Vect(A,B)$. By considering the case $A=B$, we derive the notion of a $V$-universal (co)acting bialgebra (and Hopf algebra) for a given algebra $A$. In particular, this leads to a refinement of the existence conditions for the Manin--Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the $V$-universal acting bi/Hopf algebra and the finite dual of the $V$-universal coacting bi/Hopf algebra under certain conditions on $V$ in terms of the finite topology on $\End_F(A)$., Comment: To appear in Commun. Contemp. Math

We prove the compatibility between the suspension construction and the complicial nerve of $\omega$-categories. As a motivating application, we produce a Quillen pair between the models of $(\infty,n)$-categories given by Rezk's complete Segal $\Theta_n$-spaces and Verity's $n$-complicial sets.

We establish tetrahedral symmetries of 6j-symbols for arbitrary fusion categories under minimal assumptions. As a convenient tool for our calculations we introduce the notion of a veined fusion category, which is generated by a finite set of simple objects but is larger than its skeleton. Every fusion category C contains veined fusion subcategories that are monoidally equivalent to C and which suffice to compute many categorical properties for C. The notion of a veined fusion category does not assume the presence of a pivotal structure, and thus in particular does not assume unitarity. We also exhibit the geometric origin of the algebraic statements for the 6j-symbols., 51 pages

We study the quotient of $\mathcal{T}_n = Rep(GL(n|n))$ by the tensor ideal of negligible morphisms. If we consider the full subcategory $\mathcal{T}_n^+$ of $\mathcal{T}_n$ of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category $Rep(H_n)$ where $H_n$ is a pro-reductive algebraic group. We determine the connected derived subgroup $G_n \subset H_n$ and the groups $G_{\lambda} = (H_{\lambda}^0)_{der}$ corresponding to the tannakian subcategory in $Rep(H_n)$ generated by an irreducible representation $L(\lambda)$. This gives structural information about the tensor category $Rep(GL(n|n))$, including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on $2$-torsion in $\pi_0(H_n)$., Comment: v6: changes in Section 11.7 and 11.8. v5: Replaced former lemma A.7 with Theorem B.3 in the new appendix B. v4: We proved a conjecture on determinants from an earlier version of the paper. This allowed us to extend the results on the groups $H_n$ considerably. v3: Minor changes. v2: Minor changes: Removed typos, added Corollaries 5.12, 5.13, 5.14

With representation-theoretic applications in mind, we construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. We show that reduced stratified Tate motives satisfy favorable properties including weight and t-structures. We also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory, such as Soergel-Wendt's semisimplified Hodge motives, Achar-Riche's complexes of parity sheaves, as well as Ho-Li's recent category of graded $\ell$-adic sheaves., Comment: Fixed Equation (2.1), added an assumption in Definition and Lemma 4.11, added Remarks 3.4 and 4.12. Results are unchanged

International audience; Given a crossed module $\chi$, we introduce $\chi$-graded monoidal categories and $\chi$-fusion categories. We use spherical $\chi$-fusion categories to construct (via the state sum method) 3-dimensional Homotopy Quantum Field Theories with target the classifying space $B\chi$ of the crossed module $\chi$ (which is a homotopy 2-type).

We study the question whether the representations defined by a dense subset of the unit space of a locally compact \'etale groupoid are enough to determine the reduced norm on the groupoid C$^*$-algebra. We present sufficient conditions for either conclusion, giving a complete answer when the isotropy groups are torsion-free. As an application we consider the groupoid $G(S)$ associated to a left cancellative monoid $S$ by Spielberg and formulate a sufficient condition, which we call C$^*$-regularity, for the canonical map $C^*_r(G(S))\to C^*_r(S)$ to be an isomorphism, in which case $S$ has a well-defined full semigroup C$^*$-algebra $C^*(S)=C^*(G(S))$. We give two related examples of left cancellative monoids $S$ and $T$ such that both are not finitely aligned and have non-Hausdorff associated \'etale groupoids, but $S$ is C$^*$-regular, while $T$ is not., Comment: 23 pages; v3: minor changes, more references, last section split in two v2: typos corrected, minor changes

Khovanov and Sazdanovic recently introduced symmetric monoidal categories parameterized by rational functions and given by quotients of categories of two-dimensional cobordisms. These categories generalize Deligne's interpolation categories of representations of symmetric groups. In this paper, we classify indecomposable objects and identify the associated graded Grothendieck rings of Khovanov-Sazdanovic's categories through sums of representation categories over crossed products of polynomial rings over a general field. To obtain these results, we introduce associated graded categories for Krull-Schmidt categories with filtrations as a categorification of the associated graded Grothendieck ring, and study field extensions and Galois descent for Krull-Schmidt categories., 52 pages. v2: minor fixes and clarifications

Strong Steiner $\omega$-categories are a class of $\omega$-categories that admit algebraic models in the form of chain complexes, whose formalism allows for several explicit computations. The conditions defining strong Steiner $\omega$-categories are traditionally expressed in terms of the associated chain complex, making them somewhat disconnected from the $\omega$-categorical intuition. The purpose of this paper is to characterize this class as the class of polygraphs that satisfy a loop-freeness condition that does not make explicit use of the associated chain complex and instead relies on the categorical features of $\omega$-categories., Comment: 27 pages, v2: final version

We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $\operatorname{GL}_n$, the two-parameter polynomial functors give a new interpretation of (polynomial) representations of the quantum symmetric pair $(U_{Q,q}^B(\mathfrak{gl}_n), U_q(\mathfrak{gl}_n) )$ which specializes to type AIII/AIV quantum symmetric pairs. The coideal subalgebra $U_{Q,q}^B(\mathfrak{gl}_n)$ appears in a Schur-Weyl duality with the type B Hecke algebra $\mathcal H^B_{Q,q}(d)$. We endow two-parameter polynomial functors with a cylinder braided structure which we use to construct the two-parameter Schur functors. Our polynomial functors can be precomposed with the quantum polynomial functors of type A producing new examples of action pairs., Comment: v2 corrects typos and misformulations in S8; v3 replaces S4, which contained an error; v4 the previous S8 is integrated in the rest of the paper because we can now prove proposition 2.6 (we thank Ruslan Maksimau and Catharina Stroppel for informing us of a lemma which lead to the proof). Other sections were also revised (ex S6.1, S7.3) v5: we incorporate referee suggestions throughout the paper

28 pages; International audience; This paper shows how to construct coherent presentations (presentations by generators, relations and relations among relations) of monoids admitting a right-noetherian Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of construction of coherent presentations for spherical Artin-Tits monoids: to general Artin-Tits monoids, and to Garside monoids. The result is applied to some monoids which are neither Artin-Tits nor Garside.

This paper is devoted to morphisms killing weights in a range (as defined by the first author) and to objects without these weights (as essentially defined by J. Wildeshaus) in a triangulated category endowed with a weight structure w. We describe several new criteria for morphisms and objects to satisfy these conditions. In some of them we use virtual t-truncations and a t-structure adjacent to w. In the case where the latter exists we prove that a morphism kills weights $m,...,n$ if and only if it factors through an object without these weights; we also construct new families of torsion theories and projective and injective classes. As a consequence, we obtain some "weakly functorial decompositions" of spectra (in the stable homotopy category SH) and a new description of those morphisms that act trivially on degree zero singular cohomology with coefficients in every abelian group., Comment: Several minor corrections made

We provide "growth schemes" for inductively generating uniform random $2p$-angulations of the sphere with $n$ faces, as well as uniform random simple triangulations of the sphere with $2n$ faces. In the case of $2p$-angulations, we provide a way to insert a new face at a random location in a uniform $2p$-angulation with $n$ faces in such a way that the new map is precisely a uniform $2p$-angulation with $n+1$ faces. Similarly, given a uniform simple triangulation of the sphere with $2n$ faces, we describe a way to insert two new adjacent triangles so as to obtain a uniform simple triangulation of the sphere with $2n+2$ faces. The latter is based on a new bijective presentation of simple triangulations that relies on a construction by Poulalhon and Schaeffer., Comment: New version correcting a major typo in a statement

We consider a Deligne-Mumford stack $X$ which is the quotient of an affine scheme $\operatorname{Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$., Comment: major revision, main result improved (no regularity assumption)

A $h$-instanton sheaf on a closed subscheme $X$ of some projective space endowed with an ample and globally generated line bundle $\mathcal{O}_X(h)$ is a coherent sheaf whose cohomology table has a certain prescribed shape. In this paper we deal with $h$-instanton sheaves relating them to Ulrich sheaves. Moreover, we study $h$-instanton sheaves on smooth curves and surfaces, cyclic $n$-folds, Fano $3$-folds and scrolls over arbitrary smooth curves. We also deal with a family of monads associated to $h$-instanton bundles on varieties satisfying some mild extra technical conditions., 39 pages. Final version in Journal of Pure and Applied Algebra

Whereas string diagrams for strict monoidal categories are well understood, and have found application in several fields of Computer Science, graphical formalisms for non-strict monoidal categories are far less studied. In this paper, we provide a presentation by generators and relations of string diagrams for non-strict monoidal categories, and show how this construction can handle applications in domains such as digital circuits and programming languages. We prove the correctness of our construction, which yields a novel proof of Mac Lane’s strictness theorem. This in turn leads to an elementary graphical proof of Mac Lane’s coherence theorem, and in particular allows for the inductive construction of the canonical isomorphisms in a monoidal category., LIPIcs, Vol. 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023), pages 37:1-37:19

We define, for each quasi-syntomic ring $R$ (in the sense of Bhatt-Morrow-Scholze), a category $\mathrm{DM}^{\rm adm}(R)$ of \textit{admissible prismatic Dieudonné crystals over $R$} and a natural functor from $p$-divisible groups over $R$ to $\mathrm{DM}^{\rm adm}(R)$. We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze., Replaced the notion of filtered prismatic Dieudonn\'e crystal by the equivalent (but simpler) notion of admissible prismatic Dieudonn\'e crystal; corrected and simplified (following a suggestion of Mathew) the proof of fully faithfulness; many small changes

We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for, and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau-Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Munoz's decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We will explain how these decompositions can be seen as evidence for the conjectural semiorthogonal decomposition., 32 pages; the initial arXiv submission has been split into 3 pieces and new results were added, the other 2 pieces will appear soon

We determine for which Coxeter types the associated small quotient of the 2 2 -category of Soergel bimodules is finitary and, for such a small quotient, classify the simple transitive 2 2 -representations (sometimes under the additional assumption of gradability). We also describe the underlying categories of the simple transitive 2 2 -representations. For the small quotients of general Coxeter types, we give a description for the cell 2 2 -representations.

The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic A∞ algebra equipped with a scalar product of degree −d. In particular, it acts on the cyclic Hochschild complex of any Poincare duality algebra of degree d, and that action factors through a quotient dg properad ST3−d of ribbon graphs which is in focus of this paper. We show that its cohomology properad H∙(ST3−d) is highly non-trivial and that it acts canonically on the reduced equivariant homology H¯S1∙(LM) of the loop space LM of any simply connected d-dimensional closed manifold M. By its very construction, the string topology properad H∙(ST3−d) comes equipped with a morphism from the gravity properad which is fully determined by the compactly supported cohomology of the moduli spaces Mg,n of stable algebraic curves of genus g with marked points. This result gives rise to new universal operations in string topology as well as reproduces in a unified way several known constructions: we show that (i) H∙(ST3−d) is also a properad under the properad of involutive Lie bialgebras in degree 3−d whose induced action on H¯S1∙(LM) agrees precisely with the famous purely geometric construction of M. Chas and D. Sullivan, (ii) H∙(ST3−d) is a properad under the properad of homotopy involutive Lie bialgebras in degree 2−d; (iii) E. Getzler's gravity operad injects into H∙(ST3−d) implying a purely algebraic counterpart of the geometric construction of C. Westerland establishing an action of the gravity operad on H¯S1∙(LM).

For a category B with finite products, we first characterize pseudofunctors from B to Cat whose corresponding opfibration is cartesian monoidal. Among those, we then characterize the ones which extend to pseudofunctors from internal groups to 2-groups. If B is additive, this is the case precisely when the corresponding opfibration has groupoidal fibres.

We complete the discrete cluster categories of type A as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a Hom‐finite Krull–Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom‐spaces in this new category can be described geometrically, even though the category is not 2‐Calabi–Yau and Ext‐spaces are not always symmetric. We describe all cluster‐tilting subcategories. Given such a subcategory, we define a cluster character that takes values in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite‐dimensional subrepresentations of infinite‐dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi–Yau conditions.

In this paper we will show that for a quasicompact quasiseparated scheme $X$ the hypercomplete pro-\'etale $\infty$-topos, as introduced by Bhatt and Scholze, is equivalent to the $ \infty $-category of continuous representations of the Galois category Gal$(X)$ of $X$ with values in the $\infty$-category of pyknotic spaces. In particular this proves that internally to pyknotic spaces, the hypercomplete pro-\'etale $\infty$-topos of $X$ is an $ \infty $-category of presheaves., DOCUMENTA MATHEMATICA, Vol 27 (2022), p. 2067-2106

We distinguish between faint, weak, strong and strict localizations of categories at morphism families and show that this framework captures the different types of derived functors that are considered in the literature. More precisely, we show that Kan and faint derived functors coincide when we use the classical Kan homotopy category, and when we use the Quillen homotopy category, Kan and strong derived functors coincide. Our comparison results are based on the fact that the Kan homotopy category is a weak localization and that the Quillen homotopy category is a strict localization.

Considering L as a complete residuated lattice, some further investigations on L-filter spaces are made. Firstly, it is shown that the category L-Fil of L-filter spaces is monoidal closed. Secondly, it is proved that the categories of L-Cauchy spaces and L-semi-Cauchy spaces are both bireective subcategories of L-Fil. Finally, the concept of filter-determined L-semiuniform convergence spaces is proposed and the resulting category is shown to be isomorphic to L-Fil, which can be embedded in the category of L-semiuniform convergence spaces as a bicoreective subcategory. Moreover, it is proved that L-Fil can be embedded in the category of L-semiuniform convergence spaces as a bireective subcategory whenever L is a completely distributive lattice.

In this paper, we first introduce the concept of a Rota–Baxter operator on a cocommutative weak Hopf algebra H and give some examples. We then construct Rota–Baxter operators from the normalized integral, antipode, and target map of H. Moreover, we construct a new multiplication “∗” and an antipode SB from a Rota–Baxter operator B on H such that HB=(H,∗,η,Δ,ε,SB) becomes a new weak Hopf algebra. Finally, all Rota–Baxter operators on a weak Hopf algebra of a matrix algebra are given.

In this paper some characterizations of Hom-Leibniz superalgebras are given and some of their basic properties are found. These properties can be seen as a generalization of corresponding well-known properties of Hom-Leibniz algebras. Considering the Hom-Akivis superalgebra associated to a given Hom-Leibniz superalgebra, it is observed that the Hom-super Akivis identity leads to an additional property of Hom-Leibniz superalgebras, which in turn gives a necessary and sufficient condition for Hom-super Lie admissibility of Hom-Leibniz superalgebras. We show also that every (left) Hom-Leibniz superalgebra has a natural Hom-Lie-Yamaguti superalgebra structure.

In this paper, we investigate a more general category of Θ \Theta -Yetter-Drinfeld modules ( Θ ∈ Aut H ( H ) \Theta \in {\rm{Aut}}\hspace{0.33em}H\left(H) ) over a Hom-Hopf algebra, which unifies two different definitions of Hom-Yetter-Drinfeld category introduced by Makhlouf and Panaite, Li and Ma, respectively. We show that the category of Θ \Theta -Yetter-Drinfeld modules with a bijective antipode S S is a braided tensor category and some solutions of the Hom-Yang-Baxter equation and the Yang-Baxter equation can be constructed by this category. Also by the method of symmetric pairs, we prove that if a Θ \Theta -Yetter-Drinfeld category over a Hom-Hopf algebra H H is symmetric, then H H is trivial. Finally, we find a sufficient and necessary condition for a Θ \Theta -Yetter-Drinfeld category to be pseudosymmetric.

In this paper, we describe the homotopy type of the homotopy fixed point sets of $S^{3}$-actions on rational spheres and complex projective spaces, and provide some properties of $S^{1}$-actions on a general rational complex.

We present the results of structural and magnetic properties of two types of heterostructures composed of ferroelectric BaTiO3 and dielectrics LaMnO3 or SrTiO3. By means of the first principle calculations we analyze structures of BaTiO3/LaMnO3 and BaTiO3/SrTiO3 interfaces and investigate magnetic properties and an impact of ferroelectric polarization. Experimental results of magnetic susceptibility measurements for Ba0.8Sr0.2TiO3/LaMnO3 heterostructure are presented.

When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages

We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang-Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces., Comment: 18 pages. Postprint version

We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras., 70 pages; v4 is post referee report

Nerves are frequently stretched during movement. We investigate here the effect of stretch on nerve excitability within the framework of the soliton theory. This thermodynamic theory for nerve pulse propagation relies on the presence of a melting transition in the nerve membrane. In this transition, the area of the nerve membrane and the nerve thickness change. It depends on thermodynamic variables including temperature, the chemical potentials of anesthetics and on hydrostatic pressure. A further variable relevant for movement science is the the stretching of nerves, i.e., a tension in the nerve caused by muscle contraction, the bending of joints and the pulling on extremities. We show here that the soliton theory predicts a decrease in nerve excitability upon stretching. This becomes evident in a reduction of the amplitude of compound action potentials and in the suppression of reflexes. We compare these predictions with medical findings., 8 pages, 6 figures

Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle-Guillemin-Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.

Aguiar and Ardila defined the Hopf monoid GP of generalized permutahedra and showed that it contains many submonoids that correspond to combinatorial objects. They also give a basic polynomial invariant of generalized permutahedra, which then specializes to submonoids. We define the Hopf monoid of directed graphs and show that it also embeds in GP. The resulting basic invariant coincides with the strict chromatic polynomial of Awan and Bernardi.

We introduced the notion of continuation in lambda calculus for Lambek calculus and showed that the continuation-passing style transformation could be naturally derived from the rules of Lambek calculus. Furthermore, since the answer category of a continuation is given when the whole sentence is determined, we introduced a polymorphic category and generalized the continuation-passing style transformation in the polymorphic Lambek calculus.