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This volume presents modern trends in the area of symmetries and their applications based on contributions from the workshop "Lie Theory and Its Applications in Physics", held near Varna, Bulgaria, in June 2015. Traditionally, Lie theory is a tool to build mathematical models for physical systems.Recently, the trend has been towards geometrization of the mathematical description of physical systems and objects. A geometric approach to a system yields in general some notion of symmetry, which is very helpful in understanding its structure. Geometrization and symmetries are employed in their widest sense, embracing representation theory, algebraic geometry, number theory, infinite-dimensional Lie algebras and groups, superalgebras and supergroups, groups and quantum groups, noncommutative geometry, symmetries of linear and nonlinear partial differential operators (PDO), special functions, and others. Furthermore, the necessary tools from functional analysis are included.<This is a large interdisciplinary and interrelated field, and the present volume is suitable for a broad audience of mathematicians, mathematical physicists, and theoretical physicists, including researchers and graduate students interested in Lie Theory.

Inhaltsverzeichnis

Frontmatter

Plenary Talks

Frontmatter

–Lie Algebras Canonically Associated to Probability Measures on with All Moments

In the paper Accardi et al.: Identification of the theory of orthogonal polynomials in d–indeterminates with the theory of 3–diagonal symmetric interacting Fock spaces on $$\mathbb {C} ^d$$Cd, submitted to: IDA–QP (Infinite Dimensional Anal. Quantum Probab. Related Topics), [1], it has been shown that, with the natural definitions of morphisms and isomorphisms (that will not be recalled here) the category of orthogonal polynomials in a finite number of variables is isomorphic to the category of symmetric interacting fock spaces (IFS) with a 3–diagonal structure. Any IFS is canonically associated to a $$*$$∗–Lie algebra (commutation relations) and a $$*$$∗–Jordan algebra (anti–commutation relations). In this paper we continue the study of these algebras, initiated in Accardi et al. An Information Complexity index for Probability Measures on $$\mathbb {R}$$R with all moments, submitted to: IDA–QP (Infinite Dimensional Anal. Quantum Probab. Related Topics), [2], in the case of polynomials in one variable, refine the definition of information complexity index of a probability measure on the real line, introduced there, and prove that the $$*$$∗–Lie algebra canonically associated to the probability measures of complexity index (0, K, 1), defining finite–dimensional approximations, in the sense of Jacobi sequences, of the Heisenberg algebra, coincides with the algebra of all $$K \times K$$K×K complex matrices.

Luigi Accardi, Abdessatar Barhoumi, Yun Gang Lu, Mohamed Rhaima

Special Conformal Transformations and Contact Terms

In this contribution I construct the Ward identity of special conformal transformations in momentum space and discuss some of its consequences on conformal field theory correlators. I show a few examples of covariant correlators in dimension 2 and 3 dimensions and in particular of those made of pure contact terms. I discuss in some detail the odd parity correlator in 3d and its connection with the gravitational Chern–Simons theory in 3d.

Loriano Bonora

On Nonlocal Modified Gravity and Its Cosmological Solutions

During hundred years of General Relativity (GR), many significant gravitational phenomena have been predicted and discovered. General Relativity is still the best theory of gravity. Nevertheless, some (quantum) theoretical and (astrophysical and cosmological) phenomenological difficulties of modern gravity have been motivation to search more general theory of gravity than GR. As a result, many modifications of GR have been considered. One of promising recent investigations is Nonlocal Modified Gravity. In this article we present a brief review of some nonlocal gravity models with their cosmological solutions, in which nonlocality is expressed by an analytic function of the d’Alembert-Beltrami operator $$\Box $$□. Some new results are also presented.

Ivan Dimitrijevic, Branko Dragovich, Jelena Stankovic, Alexey S. Koshelev, Zoran Rakic

Kinetics of Interface Growth: Physical Ageing and Dynamical Symmetries

Dynamical symmetries and their Lie algebra representations, relevant for the non-equilibrium kinetics of growing interfaces are discussed. Physical consequences are illustrated in the ageing of the 1D Glauber-Ising and Arcetri models.

Malte Henkel

News on SU(2|1) Supersymmetric Mechanics

We report on a recent progress in exploring the SU(2|1) supersymmetric quantum mechanics. Our focus is on the harmonic SU(2|1) superspace formalism which provides a superfield description of the multiplet $$(\mathbf{4, 4, 0})$$(4,4,0) and its “mirror” version. We present the $$\sigma $$σ-model and Wess–Zumino type actions for these multiplets, in both the superfield and the component approaches. An interesting new feature as compared to the flat $$\mathcal{N}=4, d=1$$N=4,d=1 case is the absence of the explicit SU(2|1) invariant Wess–Zumino term for the ordinary $$(\mathbf{4, 4, 0})$$(4,4,0) multiplet and yet the existence of such a term for the mirror multiplet. The superconformal subclass of the SU(2|1) invariant $$(\mathbf{4, 4, 0})$$(4,4,0) actions is also described. Its main distinguishing features are the “trigonometric” realization of the $$d=1$$d=1 conformal group SO(2, 1) and the oscillator-type potential terms in the component actions.

Evgeny Ivanov, Stepan Sidorov

Intrinsic Sound of Anti-de Sitter Manifolds

As is well-known for compact Riemann surfaces, eigenvalues of the Laplacian are distributed discretely and most of eigenvalues vary viewed as functions on the Teichmüller space. We discuss a new feature in the Lorentzian geometry, or more generally, in pseudo-Riemannian geometry. One of the distinguished features is that $$L^2$$-eigenvalues of the Laplacian may be distributed densely in $${\mathbb {R}}$$ in pseudo-Riemannian geometry. For three-dimensional anti-de Sitter manifolds, we also explain another feature proved in joint with F. Kassel [Adv. Math. 2016] that there exist countably many $$L^2$$-eigenvalues of the Laplacian that are stable under any small deformation of anti-de Sitter structure. Partially supported by Grant-in-Aid for Scientific Research (A) (25247006), Japan Society for the Promotion of Science.

Toshiyuki Kobayashi

Sphere Partition Functions and the Kähler Metric on the Conformal Manifold

We discuss marginal operators in $$\mathcal {N}=2$$N=2 Superconformal Field Theories in four dimensions. These operators are necessarily exactly marginal and they lead to a manifold, $$\mathcal {M}$$M, of Superconformal Field Theories. The space $$\mathcal {M}$$M is argued to be a Kähler manifold. We further argue that upon a stereographic projection of $$R^4$$R4 to $$S^4$$S4, the partition function $$Z_{S^4}$$ZS4 measures the Kähler potential. These results are established by a careful study of the interplay between conformal anomalies and the space $$\mathcal {M}$$M.

Efrat Gerchkovitz, Zohar Komargodski

Real Group Orbits on Flag Ind-Varieties of

We consider the complex ind-group $$G=\mathrm {SL}(\infty ,\mathbb {C})$$G=SL(∞,C) and its real forms $$G^0={\mathrm {SU}}(\infty ,\infty )$$G0=SU(∞,∞), $${\mathrm {SU}}(p,\infty )$$SU(p,∞), $$\mathrm {SL}(\infty ,\mathbb {R})$$SL(∞,R), $$\mathrm {SL}(\infty ,\mathbb {H})$$SL(∞,H). Our main object of study are the $$G^0$$G0-orbits on an ind-variety G / P for an arbitrary splitting parabolic ind-subgroup $$P\subset G$$P⊂G, under the assumption that the subgroups $$G^0\subset G$$G0⊂G and $$P\subset G$$P⊂G are aligned in a natural way. We prove that the intersection of any $$G^0$$G0-orbit on G / P with a finite-dimensional flag variety $$G_n/P_n$$Gn/Pn from a given exhaustion of G / P via $$G_n/P_n$$Gn/Pn for $$n\rightarrow \infty $$n→∞, is a single $$(G^0\cap G_n)$$(G0∩Gn)-orbit. We also characterize all ind-varieties G / P on which there are finitely many $$G^0$$G0-orbits, and provide criteria for the existence of open and closed $$G^0$$G0-orbits on G / P in the case of infinitely many $$G^0$$G0-orbits.

Mikhail V. Ignatyev, Ivan Penkov, Joseph A. Wolf

Derived Functors and Intertwining Operators for Principal Series Representations of

We consider the principal series representations $$I_\nu $$Iν induced from a character $$\nu $$ν of the upper triangular matrices B and its realization on the Frechet space of $$C^\infty $$C∞-sections of a line bundle over G / B. Its continuous dual is denoted by $$I_\nu ^*$$Iν∗. Let $$N \subset B$$N⊂B be the nilpotent subgroup whose diagonal entries are 1 and denote by $${\mathfrak n }$$n its Lie algebra. We determine $$H^0({\mathfrak n }, I_\nu ^*) $$H0(n,Iν∗) and $$H^1({\mathfrak n },I_\nu ^*)$$H1(n,Iν∗) and conclude that space of the intertwining operators $$T:I_\nu \rightarrow I_{-\nu }$$T:Iν→I-ν is 2 dimensional for some integral parameter, otherwise it is one dimensional. The intertwining operators are identified with distributions. We show that for certain parameters the support of this distribution is a point, i.e. that the intertwining operator is a differential intertwining operator.

Raul Gomez, Birgit Speh

Hyperlogarithms and Periods in Feynman Amplitudes

The role of hyperlogarithms and multiple zeta values (and their generalizations) in Feynman amplitudes is being gradually recognized since the mid 1990s. The present lecture provides a concise introduction to a fast developing subject that attracts the interests of a wide range of specialists – from number theorists to particle physicists.

Ivan Todorov

The Parastatistics Fock Space and Explicit Infinite-Dimensional Representations of the Lie Superalgebra

The defining triple relations of m pairs of parafermion operators $$f_i^\pm $$fi± and n pairs of paraboson operators $$b_j^\pm $$bj± with relative parafermion relations can be considered as defining relations for the Lie superalgebra $${\mathfrak {osp}}(2m+1|2n)$$osp(2m+1|2n) in terms of $$2m+2n$$2m+2n generators. As a consequence of this the parastatistics Fock space of order p corresponds to an infinite-dimensional unitary irreducible representation $$\mathfrak {V}(p)$$V(p) of $${\mathfrak {osp}}(2m+1|2n)$$osp(2m+1|2n), with lowest weight $$(-\frac{p}{2},\ldots ,- \frac{p}{2}|\frac{p}{2},\ldots ,\frac{p}{2})$$(-p2,…,-p2|p2,…,p2). An explicit construction of the representations $$\mathfrak {V}(p)$$V(p) is given for any m and n, as well as the computation of matrix elements of the $${\mathfrak {osp}}(2m+1|2n)$$osp(2m+1|2n) generators.

N. I. Stoilova, J. Van der Jeugt

Stepwise Square Integrable Representations: The Concept and Some Consequences

There are some new developments on Plancherel formula and growth of matrix coefficients for unitary representations of nilpotent Lie groups. These have several consequences for the geometry of weakly symmetric spaces and analysis on parabolic subgroups of real semisimple Lie groups, and to (infinite dimensional) locally nilpotent Lie groups. Many of these consequences are still under development. In this note I’ll survey a few of these new aspects of representation theory for nilpotent Lie groups and parabolic subgroups.

Joseph A. Wolf

Higher-Dimensional Unified Theories with Continuous and Fuzzy Coset Spaces as Extra Dimensions

We first briefly review the Coset Space Dimensional Reduction (CSDR) programme and present the results of the best model so far, based on the $$\mathcal {N} = 1$$N=1, $$d = 10$$d=10, $$E_8$$E8 gauge theory reduced over the nearly-Kähler manifold $$SU(3)/U(1)\times U(1)$$SU(3)/U(1)×U(1). Then, we present the adjustment of the CSDR programme in the case that the extra dimensions are considered to be fuzzy coset spaces and then, the best model constructed in this framework, too, which is the trinification GUT, $$SU(3)^3$$SU(3)3.

G. Manolakos, G. Zoupanos

String Theories and Gravity Theories

Frontmatter

Higher Genus Amplitudes in SUSY Double-Well Matrix Model for 2D IIA Superstring

We discuss a simple supersymmetric double-well matrix model which is considered to give a perturbation formulation of two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background. Full nonperturbative contributions to the free energy are computed by using the technique of random matrix theory, and the result shows that supersymmetry (SUSY) is spontaneously broken by nonperturbative effects due to instantons. In addition, one-point functions of operators that are not protected by SUSY are obtained to all orders in genus expansion.

Fumihiko Sugino

Kruskal–Penrose Formalism for Lightlike Thin-Shell Wormholes

The original formulation of the “Einstein–Rosen bridge” in the classic paper of Einstein and Rosen (1935) is historically the first example of a static spherically-symmetric wormhole solution. It is not equivalent to the concept of the dynamical and non-traversable Schwarzschild wormhole, also called “Einstein–Rosen bridge” in modern textbooks on general relativity. In previous papers of ours we have provided a mathematically correct treatment of the original “Einstein–Rosen bridge” as a traversable wormhole by showing that it requires the presence of a special kind of “exotic matter” located on the wormhole throat – a lightlike brane (the latter was overlooked in the original 1935 paper). In the present note we continue our thorough study of the original “Einstein–Rosen bridge” as a simplest example of a lightlike thin-shell wormhole by explicitly deriving its description in terms of the Kruskal–Penrose formalism for maximal analytic extension of the underlying wormhole spacetime manifold. Further, we generalize the Kruskal–Penrose description to the case of more complicated lightlike thin-shell wormholes with two throats exhibiting a remarkable property of QCD-like charge confinement.

Eduardo Guendelman, Emil Nissimov, Svetlana Pacheva, Michail Stoilov

Metric-Independent Spacetime Volume-Forms and Dark Energy/Dark Matter Unification

The method of non-Riemannian (metric-independent) spacetime volume-forms (alternative generally-covariant integration measure densities) is applied to construct a modified model of gravity coupled to a single scalar field providing an explicit unification of dark energy (as a dynamically generated cosmological constant) and dust fluid dark matter flowing along geodesics as an exact sum of two separate terms in the scalar field energy-momentum tensor. The fundamental reason for the dark species unification is the presence of a non-Riemannian volume-form in the scalar field action which both triggers the dynamical generation of the cosmological constant as well as gives rise to a hidden nonlinear Noether symmetry underlying the dust dark matter fluid nature. Upon adding appropriate perturbation breaking the hidden “dust” Noether symmetry we preserve the geodesic flow property of the dark matter while we suggest a way to get growing dark energy in the present universe’ epoch free of evolution pathologies. Also, an intrinsic relation between the above modified gravity $$+$$+ single scalar field model and a special quadratic purely kinetic “k-essence” model is established as a weak-versus-strong-coupling duality.

Eduardo Guendelman, Emil Nissimov, Svetlana Pacheva

Large Volume Supersymmetry Breaking Without Decompactification Problem

We consider heterotic string backgrounds in four-dimensional Minkowski space, where $$\mathcal{N}=1$$N=1 supersymmetry is spontaneously broken at a low scale $$m_{3/2}$$m3/2 by a stringy Scherk-Schwarz mechanism. We review how the effective gauge couplings at 1-loop may evade the “decompactification problem”, namely the proportionality of the gauge threshold corrections, with the large volume of the compact space involved in the supersymmetry breaking.

Hervé Partouche

Glueball Inflation and Gauge/Gravity Duality

We summarize our work on building glueball inflation models with the methods of the gauge/gravity duality. We review the relevant five-dimensional consistent truncation of type IIB supergravity. We consider solutions of this effective theory, whose metric has the form of a $$dS_4$$dS4 foliation over a radial direction. By turning on small (in an appropriate sense) time-dependent deformations around these solutions, one can build models of glueball inflation. We discuss a particular deformed solution, describing an ultra-slow roll inflationary regime.

Lilia Anguelova

Degenerate Metrics and Their Applications to Spacetime

The Lie groups preserving degenerate quadratic forms appear in various contexts related to spacetime. The homogeneous Galilei group is the intersection of two such groups. The structure group of sub-Riemannian geometry and of singular semi-Riemannian geometry, as well as of some submanifolds of semi-Riemannian manifolds, is of this kind. Such groups are shown to replace the Lorentz group at a very large class of singularities in general relativity. Also, these groups are shown to be fundamental in Kaluza-Klein theory and in gauge theory, where they provide an explanation why we may not be able to probe extra-dimensional lengths.

Ovidiu Cristinel Stoica

The Heun Functions and Their Applications in Astrophysics

The Heun functions are often called the hypergemeotry successors of the 21st century, because of the wide number of their applications. In this proceeding we discuss their application to the problem of perturbations of rotating and non-rotating black holes and highlight some recent results on their late-time ring-down obtained using those functions.

Denitsa Staicova, Plamen Fiziev

Integrable Systems

Frontmatter

Boundary Effects on the Supersymmetric Sine-Gordon Model Through Light-Cone Lattice Regularization

In this report, we discuss how the boundary condition of the spin-1 XXZ chain affects its low-energy effective field theory. The low-energy effective field theory of the spin-1 XXZ model is known as the supersymmetric (SUSY) sine-Gordon model. As a SUSY model, the theory consists of two subspaces called the Neveu–Schwarz (NS) sector and the Ramond (R) sector. In the Bethe-ansatz contest, the spin chain and its effective field theory are connected via the light-cone lattice regularization in the sense that these two models share the same transfer matrices. Conversely, the effective field theory is obtained in the scaling limit of the spin chain. Using the nonlinear integral equations (NLIEs) for the eigenvalues of the transfer matrices, we derived the scattering matrices of the SUSY sine-Gordon model from the large volume limit analysis of the spin-1 XXZ chain with boundary magnetic fields. At the same time, we derived the conformal dimensions of the SUSY sine-Gordon model in the small volume limit. From these quantities, we found that the different sector of the SUSY sine-Gordon model is realized from the spin-1 XXZ chain depending on the values of boundary magnetic fields.

Chihiro Matsui

Infinite Dimensional Matrix Product States for Long-Range Quantum Spin Models

We describe a systematic construction of long-range 1D and 2D SU(N) quantum spin models which is based on the algebraic structure of an underlying Wess–Zumino–Witten conformal field theory. The resulting Hamiltonians are put into the context of the Haldane-Shastry model, the paradigmatic example of long-range spin models.

Roberto Bondesan, Thomas Quella

Group Analysis of a Class of Nonlinear Kolmogorov Equations

A class of (1 + 2)-dimensional diffusion-convection equations (nonlinear Kolmogorov equations) with time-dependent coefficients is studied with Lie symmetry point of view. The complete group classification is achieved using a gauging of arbitrary elements (i.e., via reducing the number of variable coefficients) with the application of equivalence transformations. Two possible gaugings are discussed in order to show how equivalence group can serve in making the optimal choice.

Olena Vaneeva, Yuri Karadzhov, Christodoulos Sophocleous

Thermoelectric Characteristics of Parafermion Coulomb Islands

Using the explicit rational conformal field theory partition functions for the $${\mathbb Z}_k$$Zk parafermion quantum Hall states on a disk we compute numerically the thermoelectric power factor for Coulomb-blockaded islands at finite temperature. We demonstrate that the power factor is rather sensitive to the neutral degrees of freedom and could eventually be used to distinguish experimentally between different quantum Hall states having identical electric properties. This might help us to confirm whether non-Abelian quasiparticles, such as the Fibonacci anyons, are indeed present in the experimentally observed quantum Hall states.

Lachezar S. Georgiev

First Order Hamiltonian Operators of Differential-Geometric Type in 2D

We present an alternative approach to the problem of classification of first order Hamiltonian operators of differential-geometric type in 2D.

Paolo Lorenzoni, Andrea Savoldi

Exact Solutions for Generalized KdV Equations with Variable Coefficients Using the Equivalence Method

Using an example of variable-coefficient KdV equations we compare effectiveness of the “equivalence method” and the “extended mapping transformation method”. It is shown that the “equivalence method” is more efficient. A formula for generation of exact solutions for variable-coefficient KdV equations is derived.

Oksana Braginets, Olena Magda

Representation Theory

Frontmatter

Classifying Modules by Their Dirac Cohomology

This talk is a preliminary report on the joint work with Jing-Song Huang and David Vogan. The main question we address is: to what extent is an $$A_\mathfrak {q}(\lambda )$$Aq(λ) module determined by its Dirac cohomology? The focus of the talk is not so much on explaining this question and its answer, which are mentioned briefly at the end. Rather, the focus is on introducing the whole setting and giving some background material about representation theory, especially the notion of Dirac cohomology.

Pavle Pandžić

B–Orbits in Abelian Nilradicals of Types B, C and D: Towards a Conjecture of Panyushev

Let B be a Borel subgroup of a semisimple algebraic group G and let $$\mathfrak m$$m be an abelian nilradical in $$\mathfrak b=\mathrm{Lie} (B)$$b=Lie(B). Using subsets of strongly orthogonal roots in the subset of positive roots corresponding to $$\mathfrak m$$m, D. Panyushev [1] gives in particular classification of $$B-$$B-orbits in $$\mathfrak m$$m and $${\mathfrak m}^*$$m∗ and states general conjectures on the closure and dimensions of the $$B-$$B-orbits in both $$\mathfrak m$$m and $${\mathfrak m}^*$$m∗ in terms of involutions of the Weyl group. Using Pyasetskii correspondence between $$B-$$B-orbits in $$\mathfrak m$$m and $${\mathfrak m}^*$$m∗ he shows the equivalence of these two conjectures. In this Note we prove his conjecture in types $$B_n, C_n$$Bn,Cn and $$D_n$$Dn for adjoint case.

Nurit Barnea, Anna Melnikov

Anti de Sitter Holography via Sekiguchi Decomposition

In the present paper we start consideration of anti de Sitter holography in the general case of the $$(q+1)$$(q+1)-dimensional anti de Sitter bulk with boundary q-dimensional Minkowski space-time. We present the group-theoretic foundations that are necessary in our approach. Comparing what is done for $$q=3$$q=3 the new element in the present paper is the presentation of the bulk space as the homogeneous space $$G/H = SO(q,2)/SO(q,1)$$G/H=SO(q,2)/SO(q,1), which homogeneous space was studied by Sekiguchi.

Vladimir K. Dobrev, Patrick Moylan

Localization and the Canonical Commutation Relations

Let $$\mathbf{W}_n(\mathbb {R})$$Wn(R) be the Weyl algebra of index n. We have shown that by using extension and localization, it is possible to construct homomorphisms of $$\mathbf{W}_n(\mathbb {R})$$Wn(R) onto its image in a localization, or a quotient thereof, of $${\mathfrak {U}}(\mathfrak {so}(2,q))$$U(so(2,q)), the universal enveloping algebra of $$\mathfrak {so}(2,q)$$so(2,q), for n depending upon q [1]. Here we treat the $$\mathfrak {so}(2,1)$$so(2,1) case in complete detail. We establish an isomorphism of skew fields, specifically, $$\widetilde{\mathfrak {D}(\mathfrak {so}}(2,1))\simeq \mathfrak {D}_{(1,1)}(\mathbb {R})$$D(so~(2,1))≃D(1,1)(R) where $$\mathfrak {D}_{1,1}(\mathbb {R})$$D1,1(R) is the fraction field of $$\mathbf{W}_{1.1}(\mathbb {R}) \simeq \mathbf{W}_1(\mathbb {R})\otimes \mathbb {R}(y)$$W1.1(R)≃W1(R)⊗R(y) with $$\mathbb {R}(y)$$R(y) being the ring of polynomials in the indeterminate y and $$\widetilde{\mathfrak {D}(\mathfrak {so}}(2,1))$$D(so~(2,1)) is a certain extension of the skew field of fractions of $$\mathfrak {U}(\mathfrak {so}(2,1))$$U(so(2,1)), which is described below. We give applications of this result to representations. In particular we are able to construct representations of $$\mathbf{W}_1(\mathbb {R})$$W1(R) out of representations of $$\mathfrak {so}{(2,1)}$$so(2,1). Thus, we are able, for this lowest dimensional case, to obtain the canonical commutation relations and representations of them out of $$\mathfrak {so}(2,1)$$so(2,1) symmetry. Using similar results in higher dimensions [1] we are able to construct representations of $$\mathbf{W}_n(\mathbb {R})$$Wn(R) out of representations of $$\mathfrak {so}{(2,q)}$$so(2,q).

Patrick Moylan

Permutation-Symmetric Three-Body O(6) Hyperspherical Harmonics in Three Spatial Dimensions

We have constructed the three-body permutation symmetric O(6) hyperspherical harmonics which can be used to solve the non-relativistic three-body Schrödinger equation in three spatial dimensions. We label the states with eigenvalues of the $$U(1) \otimes SO(3)_{rot} \subset U(3) \subset O(6)$$U(1)⊗SO(3)rot⊂U(3)⊂O(6) chain of algebras and we present the corresponding $$K \le 4$$K≤4 harmonics. Concrete transformation properties of the harmonics are discussed in some detail.

Igor Salom, V. Dmitrašinović

Quantum Plactic and Pseudo-Plactic Algebras

We review the Robinson–Schensted–Knuth correspondence in the light of the quantum Schur–Weyl duality. The quantum plactic algebra is defined to be a Schur functor mapping a tower of left modules of Hecke algebras into a tower of $${U_q{\mathfrak {gl}}}$$Uqgl-modules. The functions on the quantum group carry a $${U_q{\mathfrak {gl}}}$$Uqgl-bimodule structure whose combinatorial spirit emerges in the RSK algorithm. The bimodule structure on the algebra of biletter words is used for a functorial formulation of the quantum pseudo-plactic algebra. The latter algebra has been proposed by Daniel Krob and Jean-Yves Thibon as a higher noncommutative analogue of the quantum torus.

Todor Popov

Conformal Invariance of the 1D Collisionless Boltzmann Equation

Dynamical symmetries of the collisionless Boltzmann transport equation, with an external driving force, are derived in $$d=1$$d=1 spatial dimensions. Both positions and velocities are considered as independent variables. The Lie algebra of dynamical symmetries is isomorphic to the 2D projective conformal algebra, but we find new non-standard representations. Several examples with explicit external forces are presented.

Stoimen Stoimenov, Malte Henkel

On Reducibility Criterions for Scalar Generalized Verma Modules Associated to Maximal Parabolic Subalgebras

In this short article we discuss reducibility criterions for scalar generalized Verma modules for maximal parabolic subalgebras of simply-laced Lie algebras.

Toshihisa Kubo

Supersymmetry and Quantum Groups

Frontmatter

On Finite W-Algebras for Lie Superalgebras in Non-Regular Case

We study finite W-algebras associated to non-regular nilpotent elements for the queer Lie superalgebra Q(n). We give an explicit presentation of the finite W-algebra for Q(3)

Elena Poletaeva

The Joseph Ideal for

Using deformation theory, Braverman and Joseph obtained an alternative characterisation of the Joseph ideal for simple Lie algebras, which included even type A. In this note we extend that characterisation to define a remarkable quadratic ideal for $$\mathfrak {sl}(m|n)$$. When $$m-n>2$$, we prove that the ideal is primitive and can also be characterised similarly to the construction of the Joseph ideal by Garfinkle.

Sigiswald Barbier, Kevin Coulembier

“Spread” Restricted Young Diagrams from a 2D WZNW Dynamical Quantum Group

The Fock representation of the Q-operator algebra for the diagonal 2D $${\widehat{su}}(n)_k\,$$su^(n)k WZNW model where $$Q=(Q_j^i), Q^i_j = a^i_\alpha \otimes \bar{a}^\alpha _j$$Q=(Qji),Qji=aαi⊗a¯jα, and $$a^i_\alpha , \bar{a}^\beta _j$$aαi,a¯jβ are the chiral WZNW “zero modes”, has a natural basis labeled by su(n) Young diagrams $$Y_\mathbf m\,$$Ym subject to the “spread” restriction $$\begin{aligned}\boxed {\mathrm{\text {spr}}\,(Y_\mathbf m):= \#(\text {columns}) +\#(\text {rows}) \le k+n =: h.}\end{aligned}$$spr(Ym):=#(columns)+#(rows)≤k+n=:h.

Ludmil Hadjiivanov, Paolo Furlan

Vertex Algebras and Lie Algebra Structure Theory

Frontmatter

Vertex Operator Algebras Associated with -Codes

We construct a vertex operator algebra associated with a $$\mathbf{Z}/k\mathbf{Z}$$Z/kZ-code of length n for an integer $$k \ge 2$$k≥2. We realize it inside a lattice vertex operator algebra as the commutant of a certain subalgebra. The vertex operator algebra is isomorphic to a known one in the cases $$k = 2,3$$k=2,3.

Tomoyuki Arakawa, Hiromichi Yamada, Hiroshi Yamauchi

Vertex Algebras in Higher Dimensions Are Homotopy Equivalent to Vertex Algebras in Two Dimensions

There is a differential graded operad associated to quadratic configuration spaces, whose class of algebras naturally contains the class of all vertex algebras. We have found that under certain shift of the degree in the cohomology these operads are isomorphic in cohomology for any even spatial dimension.

Nikolay M. Nikolov

Automorphisms of Multiloop Lie Algebras

Multiloop Lie algebras are twisted forms of classical (Chevalley) simple Lie algebras over a ring of Laurent polynomials in several variables $$k[x_1^{\pm 1},\ldots ,x_n^{\pm 1}]$$k[x1±1,…,xn±1]. These algebras occur as centreless cores of extended affine Lie algebras (EALA’s) which are higher nullity generalizations of affine Kac-Moody Lie algebras. Such a multiloop Lie algebra $$\mathcal {L}$$L, also called a Lie torus, is naturally graded by a finite root system $$\varDelta $$Δ, and thus possess a significant supply of nilpotent elements. We compute the difference between the full automorphism group of L and its subgroup generated by exponents of nilpotent elements. The answer is given in terms of Whitehead groups, also called non-stable $$K_1$$K1-functors, of simple algebraic groups over the field of iterated Laurent power series $$k((x_1))\ldots ((x_n))$$k((x1))…((xn)). As a corollary, we simplify one step in the proof of conjugacy of Cartan subalgebras in EALA’s due to Chernousov, Neher, Pianzola and Yahorau, under the assumption $$\mathrm {rank}(\varDelta )\ge 2$$rank(Δ)≥2.

Anastasia Stavrova

Contraction Admissible Pairs of Complex Six-Dimensional Nilpotent Lie Algebras

All possible pairs of complex six-dimensional nilpotent Lie algebras are considered and necessary contraction conditions are verified. The complete set of the Lie algebra couples that do not admit contraction is obtained.

Maryna Nesterenko, Severin Posta

About Filiform Lie Algebras of Order 3

The aim of this work is to review recent advances in generalizing filiform Lie (super)algebras into the theory of Lie algebras of order F. Recall that the latter type of algebras constitutes the underlying algebraic structure of fractional supersymmetry. In this context filiform Lie algebras of order F emerged in [16], and a further study can be found in [17].

R. M. Navarro

Algebraic Structures Related to Racah Doubles

In Oste and Van der Jeugt, SIGMA, 12 (2016), [13], we classified all pairs of recurrence relations connecting two sets of Hahn, dual Hahn or Racah polynomials of the same type but with different parameters. We examine the algebraic relations underlying the Racah doubles and find that for a special case of Racah doubles with specific parameters this is given by the so-called Racah algebra.

Roy Oste, Joris Van der Jeugt

A Note on Strongly Graded Lie Algebras

We show that if a strongly graded Lie algebra with a symmetric support $$({\mathfrak {L}}, [\cdot , \cdot ])$$(L,[·,·]) is centerless, then $${\mathfrak {L}}$$L is the direct sum of the family of its minimal graded-ideals, each one being a graded-simple strongly graded Lie algebra.

Antonio J. Calderón Martín, Diouf Mame Cheikh

Various Mathematical Results

Frontmatter

Toeplitz Operators with Discontinuous Symbols on the Sphere

We obtain asymptotics of norms for Toeplitz operators with specific discontinuous symbols on $$S^2$$S2.

Tatyana Barron, David Itkin

Multiplication of Distributions in Mathematical Physics

We expose a mathematical method that permits to treat calculations in form of multiplications of distributions that arise in various areas of mathematical physics, starting with an analysis of the famous Schwartz impossibility result (1954), then a construction of products of distributions, with examples and references of use in various domains of physics: classical and quantum mechanics, stochastic analysis and general relativity.

J. Aragona, P. Catuogno, J. F. Colombeau, S. O. Juriaans, Ch. Olivera

About Arbitrage and Holonomy

I make a brief survey of the realization that arbitrage in finance is holonomy of a gauge connection (I am neither an economist nor an econophysicist. Nevertheless I would like to advertise in this mathematical physics workshop a very simple but in my view beautiful and important observation. I will assume the potential reader is acquainted with elementary notions from gauge theory.).

Alexander Ganchev

On Some Exact Solutions of Heat and Mass Transfer Equations with Variable Transport Coefficients

Solutions of stationary and non-stationary heat and mass transfer equations describing thermal diffusion in a binary mixture are investigated. The dependence of physical properties on temperature and concentration is taken into account. The resulting differential equations are non–linear and require nontrivial approaches to their study and construction of exact solutions.

Ilya I. Ryzhkov, Irina V. Stepanova

A Star Product for the Volume Form Nambu-Poisson Structure on a Kähler Manifold

Every symplectic (1-plectic) manifold $$(M,\omega )$$ of dimension 2n may be regarded as a $$(2n-1)$$-plectic manifold by wedging together n copies of the symplectic 2-form to get the Liouville volume form. This volume form defines a Nambu-Poisson structure $$\{.,\dots ,.\}_{NP}$$ of order 2n on $$C^{\infty }(M)$$. When the manifold is Kähler, the Kähler structure can be used to define a star product (known as the Berezin-Toeplitz star product) for the Poisson algebra $$C^{\infty }(M)$$. For a Kähler manifold $$(M,\omega )$$ we define a higher order analogue of the Berezin-Toeplitz star product on the Nambu-Poisson algebra $$(C^{\infty }(M),\{.,\dots ,.\}_{NP})$$.

Baran Serajelahi
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