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Orthogonal abelian Cartan subalgebra decomposition of $\mathfrak{sl}_n$ over a finite commutative ringFeb 07 2018Orthogonal decomposition of the special linear Lie algebra over the complex numbers was studied in the early 1980s and attracted further attentions in the past decade due to its application in quantum information theory. In this paper, we study this decomposition ... More

Restricted One-dimensional Central Extensions of the Restricted Filiform Lie Algebras ${\frak m}_0^λ(p)$Jan 24 2018We show, for a field ${\mathbb F}$ of prime characteristic $p>0$, that the truncated filiform Lie algebra ${\frak m}_0(p)$ admits a family ${\frak m}_0^\lambda(p)$ of restricted Lie algebra structures parameterized by elements $\lambda\in {\mathbb F}^p$. ... More

Higher-order models for glioma invasion: from a two-scale description to effective equations for mass density and momentumJan 09 2018Starting from a two-scale description involving receptor binding dynamics and a kinetic transport equation for the evolution of the cell density function under velocity reorientations, we deduce macroscopic models for glioma invasion featuring partial ... More

Principal fiber bundles in non-commutative geometryDec 06 2016These are the expanded notes of a course given at the Summer school "Geometric, topological and algebraic methods for quantum field theory" held at Villa de Leyva, Colombia in July 2015. We first give an introduction to non-commutative geometry and to ... More

Compact pseudo-Riemannian homogeneous Einstein manifolds of low dimensionNov 26 2016Let $M$ be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group $G$ acts transitively and isometrically on $M$. In this situation, the metric on $M$ induces a bilinear form $\langle\cdot,\cdot\rangle$ on ... More

Compact pseudo-Riemannian homogeneous Einstein manifolds of low dimensionNov 26 2016Sep 18 2017Let $M$ be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group $G$ acts transitively and isometrically on $M$. In this situation, the metric on $M$ induces a bilinear form $\langle\cdot,\cdot\rangle$ on ... More

Schur-Weyl Duality for Heisenberg CosetsNov 01 2016Let $V$ be a simple vertex operator algebra containing a rank $n$ Heisenberg vertex algebra $H$ and let $C=\text{Com}\left( {H}, {V}\right)$ be the coset of ${H}$ in ${V}$. Assuming that the representation categories of interest are vertex tensor categories ... More

Local coordinate systems on quantum flag manifoldsOct 29 2016This paper consist of 3 sections. In the first section, we will give a brief introduction to the "Feigin's homomorphisms" and will see how they will help us to prove our main and fundamental theorems related to quantum Serre relations and screening operators. ... More

Coset construction of AdS particle dynamicsOct 26 2016We analyze dynamics of the AdS$_{N+1}$ particle realized on the coset SO$(2,N)/$SO$(1,N)$. Hamiltonian reduction provides the physical phase space in terms of the coadjoint orbit obtained by boosting a timelike element of ${\frak so}(2,N)$. We show equivalence ... More

Coset construction of AdS particle dynamicsOct 26 2016Nov 07 2016We analyze dynamics of the AdS$_{N+1}$ particle realized on the coset SO$(2,N)/$SO$(1,N)$. Hamiltonian reduction provides the physical phase space in terms of the coadjoint orbit obtained by boosting a timelike element of ${\frak so}(2,N)$. We show equivalence ... More

A robust quantitative local central limit theorem with applications to enumerative combinatorics and random combinatorial structuresOct 24 2016A useful heuristic in the understanding of large random combinatorial structures is the Arratia-Tavare principle, which describes an approximation to the joint distribution of component-sizes using independent random variables. The principle outlines ... More

Perfect and semiperfect restricted enveloping algebrasOct 20 2016For a restricted Lie algebra $L$, the conditions under which its restricted enveloping algebra $u(L)$ is semiperfect are investigated. Moreover, it is proved that $u(L)$ is left (or right) perfect if and only if $L$ is finite-dimensional.

Quantization of the ${\rm AdS}_3$ Superparticle on ${\rm OSP}(1|2)^2/{\rm SL}(2,\mathbb{R})$Oct 11 2016We analyze ${\rm AdS}_3$ superparticle dynamics on the coset ${\rm OSP}(1|2) \times {\rm OSP}(1|2)/{\rm SL}(2,\mathbb{R})$. The system is quantized in canonical coordinates obtained by gauge invariant Hamiltonian reduction. The left and right Noether ... More

Quantization of the ${\rm AdS}_3$ Superparticle on ${\rm OSP}(1|2)^2/{\rm SL}(2,\mathbb{R})$Oct 11 2016Nov 07 2016We analyze ${\rm AdS}_3$ superparticle dynamics on the coset ${\rm OSP}(1|2) \times {\rm OSP}(1|2)/{\rm SL}(2,\mathbb{R})$. The system is quantized in canonical coordinates obtained by gauge invariant Hamiltonian reduction. The left and right Noether ... More

Whittaker supports for representations of reductive groupsOct 02 2016Let $F$ be either $\mathbb{R}$ or a finite extension of $\mathbb{Q}_p$, and let $G$ be the group of $F$-points of a reductive group defined over $F$. Also let $\pi$ be a smooth representation of $G$ (Frechet of moderate growth if $F=\mathbb{R}$). For ... More

Whittaker supports for representations of reductive groupsOct 02 2016Dec 14 2016Let $F$ be either $\mathbb{R}$ or a finite extension of $\mathbb{Q}_p$, and let $G$ be a finite central extension of the group of $F$-points of a reductive group defined over $F$. Also let $\pi$ be a smooth representation of $G$ (Frechet of moderate growth ... More

Whittaker supports for representations of reductive groupsOct 02 2016Nov 11 2016Let $F$ be either $\mathbb{R}$ or a finite extension of $\mathbb{Q}_p$, and let $G$ be the group of $F$-points of a reductive group defined over $F$. Also let $\pi$ be a smooth representation of $G$ (Frechet of moderate growth if $F=\mathbb{R}$). For ... More

Finite group schemes of $p$-rank $\leq1$Sep 14 2016Let $\mathcal{G}$ be a finite group scheme over an algebraically closed field $k$ of characteristic ${\rm char}(k)=p\geq 3$. In generalization of the familiar notion from the modular representation theory of finite groups, we define the $p$-rank $\mathsf{rk}_p(\mathcal{G})$ ... More

Lie supergroups vs. super Harish-Chandra pairs: a new equivalenceSep 09 2016This paper presents a new contribution to the study of Lie supergroups - of real smooth, real analytic or complex holomorphic type - by means of super Harish-Chandra pairs. Namely, one knows that there exists a natural functor \Phi\ from Lie supergroups ... More

Gradings on modules over Lie algebras of E typesSep 08 2016For any grading by an abelian group $G$ on the exceptional simple Lie algebra $\mathcal{L}$ of type $E_6$ or $E_7$ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, ... More

Quantum exceptional group $G_2$ and its conjugacy classesSep 08 2016We construct quantization of semisimple conjugacy classes of the exceptional group $G=G_2$ along with and by means of their exact representations in highest weight modules of the quantum group $U_q(\mathfrak{g})$. With every point $t$ of a fixed maximal ... More

On structure and TKK algebras for Jordan superalgebrasSep 01 2016Sep 20 2016We compare a number of different definitions of structure algebras and TKK constructions for Jordan (super)algebras appearing in the literature. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and the TKK constructions ... More

Lie algebras with nilpotent length greater than that of each of their subalgebrasAug 24 2016The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$, and of nilpotent ... More

On the algebraic structure of Killing superalgebrasAug 21 2016We study the algebraic structure of the Killing superalgebra of a supersymmetric $11$-dimensional supergravity background and show that it is isomorphic to a filtered deformation of a $\mathbb Z$-graded subalgebra of the Poincar\'e superalgebra. We then ... More

Lie algebras and around: selected questionsAug 20 2016Several open questions are discussed. The topics include cohomology of current and related Lie algebras, algebras represented as the sum of subalgebras, structures and phenomena peculiar to characteristic $2$, and variations on themes of Ado, Whitehead, ... More

Ternary Leibniz color algebras and beyondAug 11 2016The purpose of this paper is to generalize some results on ternary Leibniz algebras to the case of ternary Leibniz color algebras. In particular, we study color Lie triple systems. In order to produce examples of ternary Leibniz color algebras from Leibniz ... More

Universal enveloping algebras and Poincaré-Birkhoff-Witt theorem for involutive Hom-Lie algebrasJul 20 2016A Hom-type algebra is called involutive if its Hom map is multiplicative and involutive. In this paper, we obtain an explicit construction of the free involutive Hom-associative algebra on a Hom-module. We then apply this construction to obtain the universal ... More

Restricted Poisson AlgebrasJul 20 2016We re-formulate Bezrukavnikov-Kaledin's definition of a restricted Poisson algebra, provide some natural and interesting examples, and discuss connections with other research topics.

On the category of finite-dimensional representations of $\OSPrn$: Part IJul 14 2016We study the combinatorics of the category F of finite-dimensional modules for the orthosymplectic Lie supergroup OSP(r|2n). In particular we present a positive counting formula for the dimension of the space of homomorphism between two projective modules. ... More

Non-commutative Geometry of Homogenized Quantum $\mathfrak{sl}(2,\mathbb{C})$Jul 02 2016This paper examines the relationship between certain non-commutative analogues of projective 3-space, $\mathbb{P}^3$, and the quantized enveloping algebras $U_q(\mathfrak{sl}_2)$. The relationship is mediated by certain non-commutative graded algebras ... More

Lie bialgebra structures on 2-step nilpotent graph algebrasJul 01 2016We generalize a result on the Heisenberg Lie algebra that gives restrictions to possible Lie bialgebra cobrackets on 2-step nilpotent algebras with some additional properties. For the class of 2-step nilpotent Lie algebras coming from graphs, we describe ... More

The fine $\spo(2|n)$-equivariant quantizations on the super circles $S^{1|n}$Jun 25 2016In this paper, we generalize the known results on the super circles $S^{1|1}$ and $S^{1|2}$. We construct the fine equivariant quantization on the super circle $S^{1|n}$ for $n\geqslant 3$. The equivariant Lie superalgebra is $\spo(2|n)$ which is constituted ... More

Twisted Coxeter elements and Folded AR-quivers via Dynkin diagram automorphisms:IIJun 01 2016Jul 22 2016As a continuation of the previous paper, we find a combinatorial interpretation of Dorey's rule for type $C_n$ via twisted Auslander-Reiten quivers (AR-quivers) of type $D_{n+1}$, which are combinatorial AR-quivers related to certain Dynkin diagram automorphisms. ... More

Twisted Coxeter elements and folded AR-quivers via Dynkin diagram automorphisms: IMay 31 2016We introduce and study the twisted adapted $r$-cluster point and its combinatorial Auslander-Reiten quivers, called twisted AR-quivers and folded AR-quivers, of type $A_{2n+1}$ which are closely related to twisted Coxeter elements and the non-trivial ... More

Twisted Coxeter elements and folded AR-quivers via Dynkin diagram automorphisms: IMay 31 2016Oct 27 2016We introduce and study the twisted adapted $r$-cluster point and its combinatorial Auslander-Reiten quivers, called twisted AR-quivers and folded AR-quivers, of type $A_{2n+1}$ which are closely related to twisted Coxeter elements and the non-trivial ... More

Invariant differential operators in positive characteristicMay 31 2016We consider an analog of the problem Veblen formulated in 1928 at the IMC: classify invariant differential operators between "natural objects" (spaces of either tensor fields, or jets, in modern terms) over a real manifold of any dimension. For unary ... More

On Superalgebras of Matrices with Symmetry PropertiesMay 27 2016It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for square matrices ... More

On extended graphical calculus for categorified quantum $sl(n)$May 22 2016We study the properties of the extended graphical calculus for categorified quantum $sl(n)$. The main results include proofs of Reidemeister 2 and Reidemeister 3-like moves involving strands corresponding to arbitrary thicknesses and arbitrary colors ... More

Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley treeMay 15 2016The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered ... More

Weak stability of the plasma-vacuum interface problemApr 13 2016May 14 2016We consider the free boundary problem for the two-dimensional plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider ... More

The spectrum on $p$-forms of a lens spaceApr 08 2016Sep 28 2016We give an explicit description of the spectrum of the Hodge--Laplace operator on $p$-forms of an arbitrary lens space for any $p$. We write the two generating functions encoding the $p$-spectrum as rational functions. As a consequence, we prove a geometric ... More

Planes in degenerate 3-manifoldsApr 07 2016We study totally geodesic planes in hyperbolic 3-manifolds $M$ having incompressible core and degenerate ends. We prove a Ratner-type phenomenon: a closed minimal $PSL(2,R)-$invariant subset of $M$ is either an immersed totally geodesic surface or all ... More

The symmetric invariants of centralizers and Slodowy grading IIApr 05 2016Aug 10 2016Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $\ell$ over an algebraically closed field $\Bbbk$ of characteristic zero, and let $(e,h,f)$ be an $\mathfrak{sl}_2$-triple of g. Denote by $\mathfrak{g}^{e}$ the centralizer of $e$ ... More

The Harish-Chandra isomorphism for quantum GL_2Mar 30 2016We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra D_q(GL_2) of quantum differential operators on GL_2, to the spherical double affine Hecke algebra associated to GL2. The isomorphism holds for all ... More

Type $D_n^{(1)}$ rigged configuration bijectionMar 26 2016We establish a bijection between the set of rigged configurations and the set of tensor products of Kirillov--Reshetikhin crystals of type $D^{(1)}_n$ in full generality. We prove the invariance of rigged configurations under the action of the combinatorial ... More

Deformation of Noncommutative Quantum MechanicsMar 18 2016In this paper, the Lie group $G_{NC}^{\alpha,\beta,\gamma}$, of which the kinematical symmetry group $G_{NC}$ of noncommutative quantum mechanics (NCQM) is a special case due to fixed nonzero $\alpha$, $\beta$ and $\gamma$, is three-parameter deformation ... More

Remarks on the ABG Induction TheoremMar 17 2016A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg compares the bounded derived category of modules for the principal block of a Lusztig quantum enveloping algebra at anroot of unity with an explicit subcategory of the bounded ... More

Elliptic Quantum Groups U_{q,p}(gl_N) and E_{q,p}(gl_N)Mar 14 2016Oct 03 2016We reformurate a central extension of Felder's elliptic quantum group in the FRST formulation as a topological algebra E_{q,p}(gl_N) over the ring of formal power series in p. We then discuss the isomorphism between E_{q,p}(gl_N) and the elliptic algebra ... More

Faithful completely reducible representations of modular Lie algebrasMar 06 2016May 19 2016The Ado-Iwasawa Theorem asserts that a finite-dimensional Lie algebra $L$ over a field $F$ has a finite-dimensional faithful module $V$. There are several extensions asserting the existence of such a module with various additional properties. In particular, ... More

Lie algebras admitting symmetric, invariant and nondegenerate bilinear formsFeb 26 2016May 30 2016We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, ... More

One-parameter deformations of the diassociative and dendriform operadsFeb 19 2016Livernet and Loday constructed a polarization of the nonsymmetric associative operad A with one operation into a symmetric operad SA with two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of SA which includes ... More

Atiyah classes and dg-Lie algebroids for matched pairsJan 23 2016Apr 01 2016For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal{M}=L[1]\oplus L/A$ such that the inclusion $\iota: A[1] \to \mathcal{M}$ and the projection $p:\mathcal{M}\to L[1]$ are morphisms ... More

A new Composition-Diamond lemma for associative conformal algebrasJan 16 2016Let $C(B,N)$ be the free associative conformal algebra generated by a set $B$ with a bounded locality $N$. Let $S$ be a subset of $C(B,N)$. A Composition-Diamond lemma for associative conformal algebras is firstly established by Bokut, Fong, and Ke in ... More

Polynomial Invariants of Torus Knots and (p,q)-CalculusJan 14 2016We introduce the deformed fermionic numbers, corresponding to the skein relations, the main characteristics of knots and links. These fermionic numbers allow one to restore the skein relations. For the Alexander (Jones) skein relation we introduce corresponding ... More

Graded simple modules and loop modulesJan 12 2016Sep 08 2016Necessary and sufficient conditions are given for a $G$-graded simple module over a unital associative algebra, graded by an abelian group $G$, to be isomorphic to a loop module of a simple module, as well as for two such loop modules to be isomorphic ... More

Quantum subgroups of simple twisted quantum groups at roots of oneJan 05 2016Sep 27 2016Let $G$ be a connected, simply connected simple complex algebraic group and let $\epsilon$ be a primitive $\ell$th root of unity with $\ell$ odd and coprime with $3$ if $G$ is of type $G_{2}$. We determine all Hopf algebra quotients of the twisted multiparameter ... More

No Lie $p$-algebras of cohomological dimension oneJan 03 2016Aug 20 2016We prove that a Lie $p$-algebra of cohomological dimension one is one-dimensional, and discuss related questions.

Regularization of Mickelsson generators for non-exceptional quantum groupsDec 29 2015Let $\mathfrak{g}'\subset \mathfrak{g}$ be the pair of Lie algebras of either symplectic or orthogonal infinitesimal endomorphisms of the complex vector spaces $\mathbb{C}^{N-2}\subset \mathbb{C}^N$ and $U_q(\mathfrak{g}')\subset U_q(\mathfrak{g})$ the ... More

Path algebras of quivers and representations of locally finite Lie algebrasDec 28 2015Jun 16 2016We explore the (noncommutative) geometry of locally simple representations of the diagonal locally finite Lie algebras $\mathfrak{sl}(n^\infty)$, $\mathfrak o(n^\infty)$, and $\mathfrak{sp}(n^\infty)$. Let $\mathfrak g_\infty$ be one of these Lie algebras, ... More

Cohomology of $\mathbb{N}$-graded Lie algebras of maximal class over $\mathbb{Z}_2$Dec 24 2015Jan 11 2016We compute the cohomology with trivial coefficients of Lie algebras $\mathfrak{m}_0$ and $\mathfrak{m}_2$ of maximal class over the field $\mathbb{Z}_2$. In the infinite-dimensional case, we show that the cohomology rings $H^*(\mathfrak{m}_0)$ and $H^*(\mathfrak{m}_2)$ ... More

A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebrasDec 17 2015Jan 23 2016Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p>0$ and suppose that $p$ is a very good prime for $G$. We prove that any maximal Lie subalgebra $M$ of $\mathfrak{g} = {\rm Lie}(G)$ with ${\rm rad}(M) \ne 0$ has ... More

Polynomial realisations of Lie (super)algebras and Bessel operatorsDec 04 2015Sep 20 2016We study realisations of Lie (super)algebras in Weyl (super)algebras and connections with minimal representations. The main result is the construction of small realisations of Lie superalgebras, which we apply for two distinct purposes. Firstly it naturally ... More

Classical invariant theory for free metabelian Lie algebrasDec 04 2015Let $KX_d$ be a vector space with basis $X_d=\{x_1,\ldots,x_d\}$ over a field $K$ of characteristic 0. One of the main topics of classical invariant theory is the study of the algebra of invariants $K[X_d]^{SL_2(K)}$, where $KX_d$ is a module of the special ... More

On tensor product decomposition of positive representations of $\mathcal{U}_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$Nov 25 2015We study the tensor product decomposition of the split real quantum group $U_{q\tilde{q}}(sl(2,R))$ from the perspective of finite dimensional representation theory of compact quantum groups. It is known that the class of positive representations of $U_{q\tilde{q}}(sl(2,R))$ ... More

Homogeneous irreducible supermanifolds and graded Lie superalgebrasNov 22 2015Oct 26 2016A depth one grading $\mathfrak{g}= \mathfrak{g}^{-1}\oplus \mathfrak{g}^0 \oplus \mathfrak{g}^1 \oplus \cdots \oplus \mathfrak{g}^{\ell}$ of a finite dimensional Lie superalgebra $\mathfrak{g}$ is called nonlinear irreducible if the isotropy representation ... More

Homogeneous irreducible supermanifolds and graded Lie superalgebrasNov 22 2015A depth one grading $\mathfrak{g}= \mathfrak{g}^{-1}\oplus \mathfrak{g}^0 \oplus \mathfrak{g}^1 \oplus \cdots \oplus \mathfrak{g}^{\ell}$ of a finite dimensional Lie superalgebra $\mathfrak{g}$ is called nonlinear irreducible if the isotropy representation ... More

Central reflections and nilpotency in exact Mal'cev categoriesNov 03 2015Jan 20 2016A general notion of nilpotency is studied in the context of exact Mal'cev categories, subsuming the notions of nilpotent group and of nilpotent Lie algebra as special cases. The nilpotent objects of class less than or equal to $n$ are shown to form a ... More

Lie identities on symmetric elements of restricted enveloping algebrasOct 29 2015Let L be a restricted Lie algebra over a field of characteristic p > 2 and denote by u(L) its restricted enveloping algebra. We determine the conditions under which the set of symmetric elements of u(L) with respect to the principal involution is Lie ... More

Lie Properties of Restricted Enveloping AlgebrasOct 29 2015Let L be a restricted Lie algebra over a field of positive characteristic. We survey the known results about the Lie structure of the restricted enveloping algebra u(L) of L. Related results about the structure of the group of units and the symmetric ... More

Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothnessOct 28 2015Aug 01 2016We consider the approximate recovery of multivariate periodic functions from a discrete set of function values taken on a rank-$s$ integration lattice. The main result is the fact that any (non-)linear reconstruction algorithm taking function values on ... More

The left-greedy Lie algebra basis and star graphsOct 23 2015We construct a basis for free Lie algebras via a ``left-greedy'' bracketing algorithm on Lyndon-Shirshov words. We use a new tool -- the configuration pairing between Lie brackets and graphs of Sinha-Walter -- to show that the left-greedy brackets form ... More

Generalized Alexander Polynomial InvariantsOct 22 2015We propose an algorithm which allows to derive the generalized Alexander polynomial invariants of knots and links with the help of the q,p-numbers, appearing in bosonic two-parameter quantum algebra. These polynomials turn into HOMFLY ones by applying ... More

Spectra of orbifolds with cyclic fundamental groupsOct 20 2015Feb 18 2016We give a simple geometric characterization of isospectral orbifolds covered by spheres, complex projective spaces and the quaternion projective line having cyclic fundamental group. The differential operators considered are Laplace-Beltrami operators ... More

Central extensions of Lie algebras of symplectic and divergence free vector fieldsOct 20 2015In this review paper, we present several results on central extensions of the Lie algebra of symplectic (Hamiltonian) vector fields, and compare them to similar results for the Lie algebra of (exact) divergence free vector fields. In particular, we comment ... More

Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebrasOct 13 2015This paper studies the concepts of a totally compatible dialgebra and a totally compatible Lie dialgebra, defined to be a vector space with two binary operations that satisfy individual and mixed associativity conditions and Lie algebra conditions respectively. ... More

Topological Rigidity ProblemsOct 11 2015Oct 15 2015We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during ... More

Graded Lie algebras of Cartan type in characteristic 2Oct 08 2015We investigate the graded Lie algebras of Cartan type $W$, $S$ and $H$ in characteristic 2 and determine their simple constituents and some exceptional isomorphisms between them. We also consider the graded Lie algebras of Cartan type $K$ in characteristic ... More

The 3x+1 problem: a lower bound hypothesisOct 06 2015Dec 15 2015Much work has been done attempting to understand the dynamic behaviour of the so-called "3x+1" function. In this paper, we formulate a new hypothesis that relates the minimal value of the first term in any finite sequence of iterations to the length of ... More

The 3x+1 problem: a lower bound hypothesisOct 06 2015Nov 17 2016Much work has been done attempting to understand the dynamic behaviour of the so-called "3x+1" function. It is known that finite sequences of iterations with a given length and a given number of odd terms have some combinatorial properties modulo powers ... More

Fractional diffusion equation with distributed-order material derivative. Stochastic foundationsOct 01 2015In this paper we present stochastic foundations of fractional dynamics driven by fractional material derivative of distributed order-type. Before stating our main result we present the stochastic scenario which underlies the dynamics given by fractional ... More

Colliding holes in Riemann surfaces and quantum cluster algebrasSep 23 2015May 24 2016We introduce the notion of bordered cusped Teichm\"uller space, as the Teichm\"uller space of Riemann surfaces with at least one hole and at least one bordered cusp on the boundary. We propose a combinatorial graph description of this bordered cusped ... More

Group gradings on the Lie and Jordan superalgebras $Q(n)$Sep 17 2015Sep 21 2015We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras $Q(n)$, $n \geq 2$, over an algebraically closed field of characteristic different from $2$ (and not dividing $n+1$ in the Lie case): fine gradings up ... More

Auslander-Reiten quiver and representation theories related to KLR-type Schur-Weyl dualitySep 16 2015Apr 25 2016We introduce new notions on the sequences of positive roots by using Auslander-Reiten quivers. Then we can prove that the new notions provide interesting information on the representation theories of KLR-algebras, quantum groups and quantum affine algebras ... More

Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneityAug 30 2015Dec 11 2015We consider a spatially-heterogeneous generalization of a well-established model for the dynamics of the Human Immunodeficiency Virus-type 1 (HIV) within a susceptible host. The model consists of a nonlinear system of three coupled reaction-diffusion ... More

On $H$-simple not necessarily associative algebrasAug 15 2015Aug 21 2015We show that if $A$ is a finite dimensional not necessarily associative $H$-simple algebra with a generalized $H$-action (e.g. $H$-module or semigroup graded) over a field of characteristic $0$, then there exists $\lim_{n\to\infty}\sqrt[n]{c_n^H(A)} \in ... More

Fischer decomposition for polynomials on superspaceAug 14 2015Recently, the Fischer decomposition for polynomials on superspace R^{m|2n} (that is, polynomials in m commuting and 2n anti-commuting variables) has been obtained unless the superdimension M=m-2n is even and non-positive. In this case, it turns out that ... More

Integrability of central extensions of the Poisson Lie algebra via prequantizationAug 04 2015May 11 2016We present a geometric construction of central S^1-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie algebra. We use this ... More

Classification of regular parametrized one-relation operadsJul 23 2015Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: \[ (a_1a_2)a_3=\sum_{\sigma\in ... More

Universal K-matrix for quantum symmetric pairsJul 22 2015Jan 29 2016Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra and let $U_q(\mathfrak{g})$ denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras $B_{c,s}$ of $U_q(\mathfrak{g})$ have ... More

On Flag Domains in the Supersymmetric SettingJul 14 2015Flag domains are open orbits of real forms $G_\mathbb{R}$ of complex reductive Lie supergroups $G$ in $G$-flag supermanifolds $Z = G/P$. This thesis discusses three topics from the theory of these flag domains: 1. Measurability(i.e. existence of $G_\mathbb{R}$-invariant ... More

Spectrum and Quantum Symmetries of the ${\rm AdS}_5 \times {\rm S}^5$ SuperstringJul 10 2015The the duality between ${\cal N}=4$ SYM and the AdS$_5\times\;$S$^5$ superstring appears to enjoy quantum integrability in the planar limit, which allowed to devise powerful methods ostensibly solving the spectral problem. However, quantization of the ... More

Spectrum and Quantum Symmetries of the ${\rm AdS}_5 \times {\rm S}^5$ SuperstringJul 10 2015Nov 07 2016The the duality between ${\cal N}=4$ SYM and the AdS$_5\times\;$S$^5$ superstring appears to enjoy quantum integrability in the planar limit, which allowed to devise powerful methods ostensibly solving the spectral problem. However, quantization of the ... More

Counterexample to a conjecture about bracesJul 08 2015Jul 23 2015We find an example of a finite solvable group (in fact, a finite $p$-group) without any left brace structure (equiv. which is not an IYB group). Our argument is an improvement of an argument of Rump, using previous work in other areas of Burde, and of ... More

Restricted One-dimensional Central Extensions of Simple Restricted Lie AlgebrasJun 30 2015Sep 25 2016In this paper, we describe restricted one-dimensional central extensions of all finite dimensional simple restricted Lie algebras defined over fields of characteristic $p\ge 5$.

Derived Equivalences of K3 Surfaces and Twined Elliptic GeneraJun 20 2015Dec 30 2015We use the unique canonically-twisted module over a certain distinguished super vertex operator algebra---the moonshine module for Conway's group---to attach a weak Jacobi form of weight zero and index one to any symplectic derived equivalence of a projective ... More

Yang-Baxter Equations, Computational Methods and ApplicationsJun 11 2015Jun 22 2015Computational methods are an important tool for solving the Yang-Baxter equations(in small dimensions), for classifying (unifying) structures, and for solving related problems. This paper is an account of some of the latest developments on the Yang-Baxter ... More

Quenched Invariance Principles for the Discrete Fourier Transforms of a Stationary ProcessMay 17 2015Jun 20 2016In this paper we study the asymptotic behavior of the normalized cadlag functions generated by the discrete Fourier transforms of a stationary square-integrable process, started at a point. We prove that the quenched invariance principle holds for averaged ... More

Affinizations and R-matrices for quiver Hecke algebrasMay 13 2015We introduce the notion of affinizations and R-matrices for arbitrary quiver Hekcke algebras. We show that they enjoy similar properties to those for symmetric quiver Hecke algebras. We next define the notion of a duality datum and construct a tensor ... More

Orbit method quantization of the AdS$_2$ superparticleApr 16 2015We consider the Hamiltonian reduction and canonical quantization of a massive AdS$_2$ superparticle realized on the coset OSP(1|2)/SO(1,1). The phase space of the massive superparticle is represented as a coadjoint orbit of a timelike element of $\mathfrak{osp}$(1|2). ... More

Orbit method quantization of the AdS$_2$ superparticleApr 16 2015Nov 07 2016We consider the Hamiltonian reduction and canonical quantization of a massive AdS$_2$ superparticle realized on the coset OSP(1|2)/SO(1,1). The phase space of the massive superparticle is represented as a coadjoint orbit of a timelike element of $\mathfrak{osp}$(1|2). ... More