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On the Lie algebra structure of $HH^1(A)$ of a finite-dimensional algebra $A$Mar 20 2019Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows that ... More
Frobenius bimodules and flat-dominant dimensionsMar 19 2019We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture on stable ... More
Transfer operators, atomic decomposition and the BestiaryMar 16 2019Arbieto and S. recently used atomic decomposition to study transfer operators. We give a long list of old and new expanding dynamical systems for which those results can be applied, obtaining the quasi-compactness of transfer operator acting on Besov ... More
Transfer operators and atomic decompositionMar 16 2019We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, ... More
The $q$-Higgs and Askey-Wilson algebrasMar 11 2019A $q$-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the $2$-sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of $\mathfrak{o}_{q^{1/2}}(4)$ ... More
Functorial PBW theorems for post-Lie algebrasMar 11 2019Using the categorical approach to Poincar\'e-Birkhoff-Witt type theorems from our previous work with Tamaroff, we prove three such theorems: for universal enveloping Rota-Baxter algebras of tridendriform algebras, for universal enveloping Rota--Baxter ... More
Quantization of continuum Kac-Moody algebrasMar 04 2019Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this paper, we prove ... More
Two series of polyhedral fundamental domains for Lorentz bi-quotientsMar 03 2019The main aim of this paper is to give two infinite series of examples of Lorentz space forms that can be obtained from Lorentz polyhedra by identification of faces. These Lorentz space forms are bi-quotients of the form $\Gamma_1\backslash G/\Gamma_2$, ... More
Smoothness of functions vs. smoothness of approximation processesMar 01 2019We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: the first based ... More
Classical simple Lie 2-algebras of toral rank 3 and a contragredient Lie 2-algebra of toral rank 4Feb 28 2019In this paper we show there are no classical type simple Lie 2-algebras with toral rank odd and we also show that the simple contragredient Lie 2-algebra $G(F_{4, a})$ of dimension 34 has toral rank 4, and we give the Cartan decomposition of $G(F_{4, ... More
An exponential lower bound for the degrees of invariants of cubic forms and tensor actionsFeb 27 2019Using the Grosshans Principle, we develop a method for proving lower bounds for the maximal degree of a system of generators of an invariant ring. This method also gives lower bounds for the maximal degree of a set of invariants that define Hilbert's ... More
Homotopy invariants for $\overline{\mathcal{M}}_{0,n}$ via Koszul dualityFeb 17 2019Feb 25 2019We show that the integral cohomology rings of the moduli spaces of stable rational marked curves are Koszul. This answers an open question of Manin. Using the machinery of Koszul spaces developed by Berglund, we compute the rational homotopy Lie algebras ... More
The Nowicki Conjecture for free metabelian Lie algebrasFeb 14 2019Let $K[X_d]=K[x_1,\ldots,x_d]$ be the polynomial algebra in $d$ variables over a field $K$ of characteristic 0. The classical theorem of Weitzenb\"ock from 1932 states that for linear locally nilpotent derivations $\delta$ (known as Weitzenb\"ock derivations) ... More
Modelling phase separation in amorphous solid dispersionsFeb 13 2019Much work has been devoted to analysing thermodynamic models for solid dispersions with a view to identifying regions in the phase diagram where amorphous phase separation or drug recrystallization can occur. However, detailed partial differential equation ... More
Existence of Kirillov--Reshetikhin crystals for multiplicity free nodesFeb 02 2019We show that the Kirillov--Reshetikhin crystal $B^{r,s}$ exists when $r$ is a node such that the Kirillov--Reshetikhin module $W^{r,s}$ has a multiplicity free classical decomposition.
Measures of weak non-compactness in preduals of von Neumann algebras and JBW$^*$-triplesJan 23 2019We prove, among other results, that three standard measures of weak non-compactness coincide in preduals of JBW$^*$-triples. This result is new even for preduals of von Neumann algebras. We further provide a characterization of JBW$^*$-triples with strongly ... More
Cohomology of Restricted Filiform Lie Algebras $\mathfrak{m}_2^λ(p)$Jan 20 2019Consider the $p$-dimensional filiform Lie algebra $\mathfrak{m}_2(p)$ over a field $\mathbb{F}$ of prime characteristic $p$ with nonzero Lie brackets $[e_1,e_i]=e_{i+1}$ for $1<i<p$ and $[e_2,e_i]=e_{i+2}$ for $2<i<p-1$. We show that there is a family ... More
The Tale of Two Categories: Inductive groupoids and Cross-connectionsJan 17 2019Jan 24 2019A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialised groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialised small category ... More
Invariant Markov semigroups on quantum homogeneous spacesJan 03 2019Invariance properties of linear functionals and linear maps on algebras of functions on quantum homogeneous spaces are studied, in particular for the special case of expected coideal *-subalgebras. Several one-to-one correspondences between such invariant ... More
Feigin and Odesskii's elliptic algebrasDec 22 2018We study the elliptic algebras $Q_{n,k}(E,\tau)$ introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. This is a family of quadratic algebras parametrized by coprime integers $n>k\geq 1$, an elliptic curve $E$, and a point $\tau\in ... More
Chiral vs classical operadDec 14 2018We establish an explicit isomorphism between the associated graded of the filtered chiral operad and the classical operad, which is useful for computing the cohomology of vertex algebras.
Pseudo Maurer-Cartan perturbation algebra and pseudo perturbation lemmaDec 14 2018We introduce the pseudo Maurer-Cartan perturbation algebra, establish a structural result and explore the structure of this algebra. That structural result entails, as a consequence, what we refer to as the pseudo perturbation lemma. This lemma, in turn, ... More
Higher polynomial identities for mutations of associative algebrasDec 13 2018We study polynomial identities satisfied by the mutation product $xpy - yqx$ on the underlying vector space of an associative algebra $A$ where $p, q$ are fixed elements of $A$. We simplify known results for identities in degree $n \le 4$, determine three ... More
KZ equations and Bethe subalgebras in generalized Yangians related to compatible R-matricesDec 12 2018The notion of compatible braidings was introduced by Isaev, Ogievetsky and Pyatov. On the base of this notion they defined certain quantum matrix algebras generalizing the RTT algebras and Reflection Equation ones. They also defined analogs of some symmetric ... More
Group gradings on the Lie and Jordan algebras of block-triangular matricesNov 14 2018We classify up to isomorphism all gradings by an arbitrary group $G$ on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic $0$. It turns out that the support of such a grading always generates ... More
On $q$- and $h$-deformations of 3d-superspaceNov 14 2018In this paper, we introduce non-standard deformations of (1+2)- and (2+1)-superspaces via a contraction using standard deformations of them. This deformed superspaces denoted by ${\mathbb A}_h^{1|2}$ and ${\mathbb A}_{h'}^{2|1}$, respectively. We find ... More
Laurent phenomenon and simple modules of quiver Hecke algebrasNov 06 2018Jan 04 2019We study consequences of a monoidal categorification of the unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category $\mathcal C_w$ strongly ... More
$SU_q(3)$ corepresentations and bivariate $q$-Krawtchouk polynomialsNov 05 2018The matrix elements of unitary $SU_q(3)$ corepresentations, which are analogues of the symmetric powers of the natural repesentation, are shown to be the bivariate $q$-Krawtchouk orthogonal polynomials, thus providing an algebraic interpretation of these ... More
Schur-Weyl duality for certain infinite dimensional $\rm{U}_q(\mathfrak{sl}_2)$-modulesNov 04 2018Jan 08 2019Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $\mathfrak{sl}_2$ at generic $q$. We show that for any $r\ge 1$, the endomorphism algebra of $M\otimes V^{\otimes r}$ is isomorphic to the type ... More
A proof of the first Kac-Weisfeiler conjecture in large characteristicsOct 30 2018Jan 28 2019In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra $\mathfrak{g}$. The first predicts the maximal dimension of simple $\mathfrak{g}$-modules and in this paper we apply the ... More
Diagram automorphisms and quantum groupsOct 10 2018Let $U^-_q = U^-_q(\mathfrak g)$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak g$, and $\sigma : \mathfrak g \to \mathfrak g$ be the automorphism obtained from the diagram automorphism. Let ... More
The twisting procedureOct 06 2018Feb 28 2019This paper provides a conceptual study of the twisting procedure, which amounts to create functorially new differential graded Lie algebras, associative algebras or operads (as well as their homotopy versions) from a Maurer--Cartan element. On the way, ... More
The twisting procedureOct 06 2018This paper provides a conceptual study of the twisting procedure, which amounts to create functorially new differential graded Lie algebras, associative algebras or operads (as well as their homotopy versions) from a Maurer--Cartan element. On the way, ... More
A Casimir element inexpressible as a Lie polynomialOct 05 2018Oct 12 2018Let $q$ be a scalar that is not a root of unity. We show that any polynomial in the Casimir element of the Fairlie-Odesskii algebra $U_q'(\mathfrak{so}_3)$ cannot be expressed in terms of only Lie algebra operations performed on the generators $I_1,I_2,I_3$ ... More
Graded polynomial identities as identities of universal algebrasOct 04 2018Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of $B$. Then $A$ ... More
Characterization of queer supercrystalsSep 12 2018We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al.. This characterization is a combination of local queer axioms generalizing Stembridge's local axioms for crystal bases for ... More
The Laurent extension of quantum plane: a complete list of symmetriesSep 09 2018This work finishes a classification of $U_q(sl_2)$-symmetries on the Laurent extension $C_q[x^{\pm 1},y^{\pm 1}]$ of quantum plane. After reproducing the partial results of a previous paper of the author related to symmetries with non-trivial action of ... More
Crystal bases and categorificationsSep 01 2018Sep 08 2018This is a survey paper of the theory of crystal bases, global bases and the cluster algebra structure on the quantum coordinate rings.
Semi-infinite highest weight categoriesAug 24 2018We develop the axiomatics of highest weight (and various more general stratified) categories, in order to incorporate two "semi-infinite" situations which are in Ringel duality with each other; the underlying posets are either upper finite or lower finite. ... More
The Nowicki Conjecture for relatively free algebrasAug 15 2018A linear locally nilpotent derivation of the polynomial algebra $K[X_m]$ in $m$ variables, over a field $K$ of characteristic 0, is called a Weitzenb\"ock derivation. It is well known from the classical theorem of Weitzenb\"ock that the algebra of constants ... More
Generalized and degenerate Whittaker quotients and Fourier coefficientsJul 30 2018The study of Whittaker models for representations of reductive groups over local and global fields has become a central tool in representation theory and the theory of automorphic forms. However, only generic representations have Whittaker models. In ... More
Product formula for the limits of normalized characters of Kirillov-Reshetikhin modulesJul 30 2018The normalized characters of Kirillov-Reshetikhin modules over a quantum affine algebra have a limit as a formal power series. Mukhin and Young found a conjectural product formula for this limit, which resembles the Weyl denominator formula. We prove ... More
The coefficients in the quantum Cayley-Hamilton formula for the reflection equation algebra expressed by a quantum Cauchy-Binet formulaJul 23 2018The quantum Cayley-Hamilton theorem for the generator of the reflection equation algebra has been proven by Pyatov and Saponov and formulas for the coefficients $\sigma_{q}(i)$ in the Cayley-Hamilton formula are given. However, these formulas do not give ... More
Operads and Maurer-Cartan spacesJul 05 2018This thesis is divided into two parts. The first one is composed of recollections on operad theory, model categories, simplicial homotopy theory, rational homotopy theory, Maurer-Cartan spaces, and deformation theory. The second part deals with the theory ... More
Differential properties of spaces of symmetric real matricesJul 03 2018We study the differential geometric properties of the manifold of non-singular symmetric real matrices endowed with the trace metric; in case of positive definite matrices we describe the full group of isometries
Graded-simple algebras and cocycle twisted loop algebrasJul 03 2018Oct 26 2018The loop algebra construction by Allison, Berman, Faulkner, and Pianzola, describes graded-central-simple algebras with split centroid in terms of central simple algebras graded by a quotient of the original grading group. Here the restriction on the ... More
Drinfeld Yangian of the queer Lie superalgebra. IJul 02 2018Jul 15 2018Drinfeld Yangian of a queer Lie superalgebra is defined as the quantization of a Lie bisuperelgebra of twisted polynomial currents. An analogue of the new system of generators of Drinfeld is being constructed. It is proved for the partial case Lie superalgebra ... More
Higher Deformations of Lie Algebra Representations IJul 02 2018Aug 28 2018In the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory of these algebras ... More
Optimal Assignments with SupervisionsJul 02 2018In this paper we provide a new graph theoretic proof of the tropical Jacobi identity, recently obtained in [AGN16]. We also develop an application of this theorem to optimal assignments with supervisions. That is, optimally assigning multiple tasks to ... More
Special Identities for Comtrans AlgebrasJun 26 2018Oct 08 2018Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Gr\"obner basis for the comtrans operad, and state a conjecture on its dimension formula. We study multilinear polynomial identities for ... More
On Jordan doubles of slow growth of Lie superalgebrasJun 26 2018Mar 13 2019To an arbitrary Lie superalgebra $L$ we associate its Jordan double ${\mathcal Jor}(L)$, which is a Jordan superalgebra. This notion was introduced by the second author before. Now we study further applications of this construction. First, we show that ... More
On Jordan doubles of slow growth of Lie superalgebrasJun 26 2018To an arbitrary Lie superalgebra $L$ we associate its Jordan double ${\mathcal Jor}(L)$, which is a Jordan superalgebra. This notion was introduced by the second author before. Now we study further applications of this construction. First, we show that ... More
An operadic approach to vertex algebra and Poisson vertex algebra cohomologyJun 22 2018We translate the construction of the chiral operad by Beilinson and Drinfeld to the purely algebraic language of vertex algebras. Consequently, the general construction of a cohomology complex associated to a linear operad produces a vertex algebra cohomology ... More
Gradings on simple real Lie algebrasJun 14 2018English: This work is a doctoral thesis in mathematics by compendium of four articles. Here we explain, using a language as simple as possible, the results achieved in those articles. The general objective is the classification of gradings on simple real ... More
Weak Algebra Bundles and Associator VarietiesMay 24 2018Nov 22 2018Algebra bundles, in the strict sense, appear in many areas of geometry and physics. However, the structure of an algebra is flexible enough to vary non-trivially over a connected base, giving rise to a structure of a weak algebra bundle. We will show ... More
Poisson Cohomology of Broken Lefschetz FibrationsMay 11 2018We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold singularities and singular points. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology ... More
Hom-Lie structures on Kac-Moody algebrasMay 01 2018Jul 29 2018We describe Hom-Lie structures on affine Kac-Moody and related Lie algebras, and discuss the question when they form a Jordan algebra.
Fractal nil graded Lie, associative, Poisson, and Jordan superalgebrasApr 20 2018We construct a just infinite fractal 3-generated Lie superalgebra $\mathbf Q$ over arbitrary field, which gives rise to an associative hull $\mathbf A$, a Poisson superalgebra $\mathbf P$, and two Jordan superalgebras $\mathbf J$, $\mathbf K$. One has ... More
Isomorphism between super Yangian and quantum loop superalgebra. IApr 18 2018Following V. Toledano-Laredo and S. Gautam approach we construct isomorphism between super $\hbar$-Yangian $Y_{\hbar}(A(m,n))$ of special linear superalgebra and quantum loop superalgebra $U_{\hbar}(LA(m,n))$.
Endofunctors and Poincaré-Birkhoff-Witt theoremsApr 17 2018Sep 03 2018We determine what appears to be the bare-bones categorical framework for Poincar\'e-Birkhoff-Witt type theorems about universal enveloping algebras of various algebraic structures. Our language is that of endofunctors; we establish that a natural transformation ... More
Mitigating Docker Security IssuesApr 13 2018It is very easy to run applications in Docker. Docker offers an ecosystem that offers a platform for application packaging, distributing and managing within containers. However, Docker platform is yet not matured. Presently, Docker is less secured as ... More
Factorisation of quasi K-matrices for quantum symmetric pairsApr 09 2018The theory of quantum symmetric pairs provides a universal K-matrix which is an analogue of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal K-matrix is a quasi K-matrix which has so far only been constructed ... More
Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear RegressionApr 08 2018Jun 29 2018Concentration inequalities form an essential toolkit in the study of high-dimensional statistical methods. Most of the relevant statistics literature is based on the assumptions of sub-Gaussian/sub-exponential random vectors. In this paper, we bring together ... More
Identities and isomorphisms of finite-dimensional graded simple algebrasApr 04 2018Jun 07 2018Let $\mathbb F$ be an algebraically closed field, $G$ be an abelian group, and let $A$ and $B$ be arbitrary finite-dimensional $G$-graded simple algebras over $\mathbb F$. We prove that $A$ and $B$ are isomorphic if, and only if, they satisfy the same ... More
Gradings on semisimple algebrasMar 27 2018Sep 05 2018The classification of gradings by abelian groups on finite direct sums of simple finite-dimensional nonassociative algebras over an algebraically closed field is reduced, by means of the use of loop algebras, to the corresponding problem for simple algebras. ... More
Deformations and their controlling cohomologies of $\mathcal{O}$-operatorsMar 25 2018$\mathcal{O}$-operators are important in broad areas in mathematics and physics, such as integrable systems, the classical Yang-Baxter equation, pre-Lie algebras and splitting of operads. In this paper, a deformation theory of $\mathcal{O}$-operators ... More
Tetrahedron Equation and Quantum $R$ Matrices for $q$-Oscillator Representations Mixing Particles and HolesMar 05 2018Jul 04 2018We construct $2^n+1$ solutions to the Yang-Baxter equation associated with the quantum affine algebras $U_q\big(A^{(1)}_{n-1}\big)$, $U_q\big(A^{(2)}_{2n}\big)$, $U_q\big(C^{(1)}_n\big)$ and $U_q\big(D^{(2)}_{n+1}\big)$. They act on the Fock spaces of ... More
On some properties of the functors ${\mathcal F}^G_P$ from Lie algebra to locally analytic representationsFeb 21 2018For a split reductive group $G$ over a finite extension $L$ of ${\mathbb Q}_p$, and a parabolic subgroup $P \subset G$ we examine functorial properties of the functors ${\mathcal F}^G_P$ introduced in \cite{OS2,OS3}. We discuss the aspects of faithfulness, ... More
Endomorphisms of Koszul complexes: formality and application to deformation theoryFeb 20 2018Dec 16 2018We study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative $K$-algebra $R$ and we prove that it is homotopy abelian over $K$, while it is generally not formal over $R$. We apply ... More
Approximation of multivariate periodic functions based on sampling along multiple rank-1 latticesFeb 19 2018In this work, we consider the approximate reconstruction of high-dimensional periodic functions based on sampling values. As sampling schemes, we utilize so-called reconstructing multiple rank-1 lattices, which combine several preferable properties such ... More
New Reductions of a Matrix Generalized Heisenberg Ferromagnet EquationFeb 10 2018We present in this report 1+1 dimensional nonlinear partial differential equation integrable through inverse scattering transform. The integrable system under consideration is a pseudo-Hermitian reduction of a matrix generalization of classical 1+1 dimensional ... More
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
2-reflective modular forms: A Jacobi forms approachJan 29 2018Mar 12 2018We give an explicit formula to express the weight of $2$-reflective modular forms. We prove that there is no $2$-reflective lattice of signature $(2,n)$ when $n\geq 15$ and $n\neq 19$ except the even unimodular lattices of signature $(2,18)$ and $(2,26)$. ... More
Weyl invariant $E_8$ Jacobi formsJan 25 2018Nov 11 2018We investigate $W(E_8)$-invariant Jacobi forms which are the Jacobi forms invariant under the action of the Weyl group of the root system $E_8$. This type of Jacobi forms has applications in mathematics and physics, but very little has been known about ... 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
Restricted One-dimensional Central Extensions of the Restricted Filiform Lie Algebras ${\frak m}_0^λ(p)$Jan 24 2018Aug 22 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
Monoidal categorification of cluster algebras (merged version)Jan 16 2018We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$, associated with a symmetric Kac-Moody algebra and its Weyl group element $w$, admits a monoidal categorification via the representations ... 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
Higher-order models for glioma invasion: from a two-scale description to effective equations for mass density and momentumJan 09 2018Apr 03 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
Admissibility and the $C_2$ SpiderJan 03 2018A tensor category is multiplicity-free if for any objects $A,B,C$ we have that $\mathrm{Hom}(A\otimes B\otimes C,\mathbb{C})$ is either $0$ or $1$ dimensional. It is known that $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ is not multiplicty-free. We find a full ... More
Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2Dec 04 2017Aug 10 2018Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set $\lambda=\Lambda_{1} - \Lambda_{2}$, where $\Lambda_{1}$, $\Lambda_{2}$ are the fundamental weights. Denote by $V(\lambda)$ the extremal weight module of extremal weight $\lambda$ ... More
Existence of Kirillov-Reshetikhin crystals of type $G_2^{(1)}$ and $D_4^{(3)}$Oct 31 2017Jul 24 2018In this paper we prove that every Kirillov-Reshetikhin module of type $G_2^{(1)}$ and $D_4^{(3)}$ has a crystal pseudobase (crystal base modulo signs), by applying the criterion for the existence of a crystal pseudobase due to Kang et al.
A classification of small operators using graph theoryOct 21 2017Given a real $n \times m$ matrix $B$, its operator norm can be defined as $$|B|=\max_{|v|=1}|Bv|.$$ We consider a matrix "small" if it has non-negative integer entries and its operator norm is less than $2$. These matrices correspond to bipartite graphs ... More
A minimal representation of the orthosymplectic Lie supergroupOct 19 2017Sep 03 2018We construct a minimal representation of the orthosymplectic Lie supergroup $OSp(p,q|2n)$, generalising the Schr\"odinger model of the minimal representation of $O(p,q)$ to the super case. The underlying Lie algebra representation is realized on functions ... More
Monoidal categories of modules over quantum affine algebras of type A and BOct 18 2017We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules over the quantum ... More
Jantzen filtration and strong linkage principle for modular Lie superalgebrasOct 16 2017In this paper, we introduce super Weyl groups, their distinguished elements and properties for basic classical Lie superalgebras. Then we formulate Jantzen filtration for baby Verma modules in graded restricted module categories of basic classical Lie ... More
Products of CW complexesOct 15 2017Aug 30 2018CW complexes are used extensively in algebraic topology, but the product of two CW complexes need not be a CW complex, as shown by Dowker. Whilst Whitehead and Milnor gave sufficient conditions for the product to be a CW complex, all existing characterizations ... More
The equation solvability problem over nilpotent Mal'cev algebrasOct 09 2017May 14 2018By a result of Horv\'ath the equation solvability problem over finite nilpotent groups and rings is in P. We generalize his result, showing that the equation solvability over every finite supernilpotent Mal'cev algebra is in P. We also give an example ... More
Drinfeld-Sokolov reduction in quantum algebrasOct 04 2017Applying the method of the paper [CT], we perform a quantum version of the Drinfeld-Sokolov reduction in Reflection Equation algebras and braided Yangians, associated with involutive and Hecke symmetries of general forms. This reduction is based on the ... More
Young wall model for $A_2^{(2)}$-type adjoint crystalsSep 29 2017Oct 12 2017We construct a Young wall model for higher level $A_2^{(2)}$-type adjoint crystals. The Young walls and reduced Young walls are defined in connection with affin energy function. We prove that the affine crystal consisiting of reduced Young walls provides ... More
The center of the reflection equation algebra via quantum minorsSep 26 2017We give simple formulas for the elements $c_k$ appearing in a quantum Cayley-Hamilton formula for the reflection equation algebra (REA) associated to the quantum group $U_q(\mathfrak{gl}_N)$, answering a question of Kolb and Stokman. The $c_k$'s are certain ... More
Kostant-Lusztig $\mathbb A$-bases of Multiparameter Quantum GroupsSep 25 2017Jan 19 2018We study the Kostant-Lusztig $\mathbb A$-base of the multiparameter quantum groups. To simplify calculations, especially for $G_2$-type, we utilize the duality of the pairing of the universal $R$-matrix.
Lie polynomials in $q$-deformed Heisenberg algebrasSep 08 2017Nov 15 2017Let $\mathbb{F}$ be a field, and let $q\in\mathbb{F}$. The $q$-deformed Heisenberg algebra is the unital associative $\mathbb{F}$-algebra $\mathcal{H}(q)$ with generators $A,B$ and relation $AB-qBA=I$, where $I$ is the multiplicative identity in $\mathcal{H}(q)$. ... More
Monomial bases and pre-Lie structure for free Lie algebrasAug 28 2017In this paper, we construct a pre-Lie structure on the free Lie algebra L(E) generated by a set E, giving an explicit presentation of L(E) as the quotient of the free pre-Lie algebra generated by E, by some ideal I. The main result in this paper is a ... More
Jordan Matsuo algebras over fields of characteristic 3Aug 07 2017The Matsuo algebra associated with a connected Fischer space is shown to be a Jordan algebra over a field of characteristic 3 if and only if the Fischer space is isomorphic to either the affine space of order 3 or the Fischer space associated with the ... More
On dimension growth of modular irreducible representations of semisimple Lie algebrasAug 04 2017Jun 27 2018In this paper we investigate the growth with respect to $p$ of dimensions of irreducible representations of a semisimple Lie algebra $\mathfrak{g}$ over $\overline{\mathbb{F}}_p$. More precisely, it is known that for $p\gg 0$, the irreducibles with a ... More
On Finite dimensional Nichols algebras of diagonal typeJul 26 2017This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, ... More
Gradings on classical central simple real Lie algebrasJul 25 2017For any abelian group $G$, we classify up to isomorphism all $G$-gradings on the classical central simple Lie algebras, except those of type $D_4$, over the field of real numbers (or any real closed field).
Parameter identification via optimal control for a Cahn--Hilliard-chemotaxis system with a variable mobilityJul 21 2017We consider the inverse problem of identifying parameters in a variant of the diffuse interface model for tumour growth model proposed by Garcke, Lam, Sitka and Styles (Math. Models Methods Appl. Sci. 2016). The model contains three constant parameters; ... More
Fractal just infinite nil Lie superalgebra of finite widthJul 20 2017Feb 12 2018The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. Their natural analogues are self-similar nil Lie $p$-algebras. In characteristic zero, similar examples of Lie algebras do not exist (Martinez and Zelmanov). The second ... More