A notion of quantum matrix (QM-) algebra generalizes and unifies two famous families of algebras from the theory of quantum groups: the RTT-algebras and the reflection equation (RE-) algebras. These algebras being generated by the components of a ‘quantum’ matrix M possess certain properties which resemble structure theorems of the ordinary matrix theory. It turns out that such structure results are

We consider the 4D Martínez Alonso-Shabat equation E uty=uzuxy−uyuxz (also referred to as the universal hierarchy equation) and using its known Lax pair construct two infinite-dimensional differential coverings over E. In these coverings, we give a complete description of the Lie algebras of nonlocal symmetries. In particular, our results generalize the ones obtained in Morozov and Sergyeyev (2014)

We derive formulas for the mean curvature of special Lagrangian 3-folds in the general case where the ambient 6-manifold has intrinsic torsion. Consequently, we are able to characterize those SU(3)-structures for which every special Lagrangian 3-fold is a minimal submanifold. In the process, we obtain an obstruction to the local existence of special Lagrangian 3-folds.

Quantization of the Teichmüller space of a non-compact Riemann surface has emerged in 1980s as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston’s shear coordinate functions on the edges form a coordinate system for the Teichmüller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon a change

We obtain new restrictions on Maslov classes of monotone Lagrangian submanifolds of ℂn. We also construct families of new examples of monotone Lagrangian submanifolds, which show that the restrictions on Maslov classes are sharp in certain cases.

Using the relativistic Fermat’s principle, we establish a bridge between stationary-complete manifolds which satisfy the observer-manifold condition and pre-Randers metrics, namely, Randers metrics without any restriction on the one-form. As a consequence, we give a description of the causal ladder of such spacetimes in terms of the elements associated with the pre-Randers metric: its geodesics and

It is shown that the normalized trace of Fedosov star product for quantum moment map depends only on the path component in the cohomology class of the symplectic form and the cohomology class of the closed formal 2-form required to define Fedosov connections (Theorem 1.3). As an application we obtain a family of obstructions to the existence of closed Fedosov star products naturally attached to symplectic

The main aim of this paper is to investigate the nature of invariancy of rectifying curve under conformal transformation and obtain a sufficient condition for which such a curve remains conformally invariant. It is shown that the normal component and the geodesic curvature of the rectifying curve is homothetic invariant.

We study anticommutative algebras with the property that commutator of any two multiplications is a derivation.

E. Cartan’s method of moving frames is applied to 3-dimensional manifolds M which are CR-embedded in 5-dimensional real hyperquadrics Q in order to classify M up to CR symmetries of Q given by the action of one of the Lie groups SU(3,1) or SU(2,2). In the latter case, the CR structure of M derives from a shear-free null geodesic congruence on Minkowski spacetime, and the relationship to relativity

We examine symmetry breaking in field theory within the framework of derived geometry, as applied to field theory via the Batalin–Vilkovisky formalism. Our emphasis is on the standard examples of Ginzburg–Landau and Yang–Mills–Higgs theories and is primarily interpretive. The rich, sophisticated language of derived geometry captures the physical story elegantly, allowing for sharp formulations of slogans

The equations of Löwner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric conformal maps and the other one is the theory of integrable systems. In this paper we compare the both approaches. After recalling the derivation of Löwner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable

There are two well-known ways of describing elements of the rotation group SO(m). First, according to the Cartan-Dieudonné theorem, every rotation matrix can be written as an even number of reflections. And second, they can also be expressed as the exponential of some anti-symmetric matrix. In this paper, we study similar descriptions of a group of rotations SO0 in the superspace setting. This group

A novel approach to the connection theory on a differential manifold is presented here. In this framework, the notion of a kth order (k=1,2,…) curvature tensor arises in a natural way. For k=1 our curvature tensor is equivalent to the conventional Riemann tensor. This approach provides natural tools for analysis of higher order variational problems in General Relativity Theory.

Alexei Kotov and Thomas Strobl have introduced a covariantized formulation of Yang–Mills-Higgs gauge theories whose main motivation was to replace the Lie algebra with Lie algebroids. This allows the introduction of a possibly non-flat connection ∇ on this bundle, after also introducing an additional 2-form ζ in the field strength. We will study this theory in the simplified situation of Lie algebra

In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have showed that a four dimensional gradient shrinking Ricci soliton satisfying some conditions

We study the convergence aspects of the metric on spectral truncations of geometry. We find general conditions on sequences of operator system spectral triples that allows one to prove a result on Gromov–Hausdorff convergence of the corresponding state spaces when equipped with Connes’ distance formula. We exemplify this result for spectral truncations of the circle, Fourier series on the circle with

The purpose in this paper is to give the structure theorem for Vaisman solvmanifolds. Moreover, we prove that a Vaisman solvmanifold has no non-Vaisman LCK structures.

In this paper, we estimate the curvature of the ρ Einstein soliton with 0≤ρ<12(n−1). We show that the curvature operator is at most polynomial growth if the Ricci curvature is bounded and the volume of the unit ball has a uniform lower bound. Furthermore, for 4 dimensional ρ Einstein soliton, the curvature operator is bounded if the Ricci curvature is bounded.

Bryant–Salamon constructed three 1-parameter families of complete manifolds with holonomy G2 which are asymptotically conical to a holonomy G2 cone. For each of these families, including their asymptotic cone, we construct a fibration by asymptotically conical and conically singular coassociative 4-folds. We show that these fibrations are natural generalizations of the following three well-known coassociative

The real Jacobi group GnJ(R), defined as the semidirect product of the Heisenberg group Hn(R) with the symplectic group Sp(n,R), admits a matrix embedding in Sp(n+1,R). The modified pre-Iwasawa decomposition of Sp(n,R) allows us to introduce a convenient coordinatization Sn of GnJ(R), which for G1J(R) coincides with the S-coordinates. Invariant one-forms on GnJ(R) are determined. The formula of the

We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized Kähler structures on maximal flag manifolds under B-transformations. We give an alternative description of the moduli space of generalized complex structures using pure spinors, and describe a cell decomposition of these moduli spaces induced by the action of the Weyl group.

The Geometrization of Quantum Mechanics proposed in this work is based on the postulate that the quantum probability density can curve the classical spacetime. It is shown that the gravitational field produced by smearing a point-mass Mo at r=0 throughout all of space (in a spherically symmetric fashion) can be interpreted as the gravitational field generated by a self-gravitating anisotropic fluid

Periodic instantons, also called calorons, are the BPS solutions to the pure Yang–Mills theories on R3×S1. It is known that the calorons interconnect with the instantons and the BPS monopoles as the ratio of their size to the period of S1 varies. We give, in this paper, the action density configurations of the SU(2) calorons of higher instanton charges with several platonic symmetries through the numerical

We show that countable metric spaces always have quantum isometry groups, thus extending the class of metric spaces known to possess such universal quantum-group actions. Motivated by this existence problem we define and study the notion of loose embeddability of a metric space (X,dX) into another, (Y,dY): the existence of an injective continuous map that preserves both equalities and inequalities

We show a coincidence of index of rigidity of differential equations with irregular singularities on a compact Riemann surface and Euler characteristic of the associated spectral curves which are recently called irregular spectral curves. Also we present a comparison of local invariants, so called Milnor formula which links the Komatsu–Malgrange irregularity of differential equations and Milnor number

We set the goal to study the properties of perfect fluid spacetimes endowed with the gradient η-Ricci and gradient Einstein solitons.

Following Boalch–Yamakawa, Li-Bland–Ševera and Meinrenken, we consider a certain class of moduli spaces on bordered surfaces from a quasi-Hamiltonian perspective. For a given Lie group G, these character varieties parametrize flat G-connections on “twisted” local systems, in the sense that the transition functions take values in G⋊Aut(G). After reviewing the necessary tools to discuss twisted quasi-Hamiltonian

I construct a global version of the local polysymplectic approach to covariant Hamiltonian field theory pioneered by C. Günther. Beginning with the geometric framework of the theory, I specialize to vertical vector fields to construct the (poly)symplectic structures, derive Hamilton’s field equations, and construct a generalized Poisson structure. I then examine a few key examples to determine the

In this paper, we study a class of Zd-graded modules, which are constructed using Shen–Larsson’s functor from sld-modules V, for the Lie algebras of divergence zero vector fields on tori and quantum tori. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with

We study BiHom–Novikov–Poisson algebras, which are twisted generalizations of Novikov–Poisson algebras and Hom–Novikov–Poisson algebras, and find that BiHom–Novikov–Poisson algebras are closed under tensor products and several kinds of perturbations. Necessary and sufficient conditions are given under which BiHom–Novikov–Poisson algebras give rise to BiHom-Poisson algebras.

In this paper, we define a quaternionic operator whose scalar part is a real parameter and vector part is a curve in three dimensional real vector space R3. We prove that quaternion product of this operator and a spherical curve represents a ruled surface in R3 if the vector part of the quaternionic operator is perpendicular to the position vector of the spherical curve. We express this surface as

We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider first-order partial differential operators ∇i± forming the Lie algebra [∇j±,∇k±]=iRjk± and [∇j+,∇k−]=i12(Rjk++Rjk−) with some anti-symmetric matrices Rij± and define the corresponding Laplacians Δ±=g±ij∇i±∇j± with some positive matrices g±ij. We show that the heat semigroups exp(tΔ±) can be represented as a Gaussian average

We degenerate the finite gap solutions of the KdV equation from the general formulation given in terms of abelian functions when the gaps tend to points, to get solutions to the KdV equation given in terms of Fredholm determinants and wronskians. For this we establish a link between Riemann theta functions, Fredholm determinants and wronskians. This gives the bridge between the algebro-geometric approach

We consider a billiard problem for compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We show that there are two types of confocal families in such setting. Using an algebro-geometric integration technique, we prove that the billiard within generalized ellipses of each type is integrable in the sense of Liouville. Further, we prove a generalization of the

Given a vector bundle A→M we study the geometry of the graded manifolds T∗[k]A[1], including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical structures, such as higher Courant algebroids on A⊕⋀k−1A∗ and higher Dirac structures therein, semi-direct products of Lie algebroid structures on A with their coadjoint

The lift of K-theoretic D-brane charge to M-theory was recently hypothesized to land in Cohomotopy cohomology theory. To further check this Hypothesis H, here we explicitly compute the constraints on fractional D-brane charges at ADE-orientifold singularities imposed by the existence of lifts from equivariant K-theory to equivariant Cohomotopy theory, through Boardman’s comparison homomorphism. We

We construct spaces of 1-dimensional supersymmetric Euclidean field theories and show that they represent real or complex K-theory. This is done without prescribing the space of states in the definition of field theory.

What will be if, given a pure stationary state on a compact hyperbolic surface, we start applying raising operator every ħ ”adiabatic” second? It turns that during adiabatic time comparable to 1 wavefunction will change as a wave traveling with a finite speed (with respect to the adiabatic time), whereas the semiclassical measure of the system will undergo a controllable transformation possessing a

In this paper, we establish optimal inequalities involving generalized δ-Casorati curvature δC(k;s−1) for the bi-slant submanifolds of generalized Sasakian space forms endowed with a quarter-symmetric connection. The equality case is discussed for ideal submanifolds. Furthermore, the special cases of derived inequality is given for C-totally real submanifolds and some other classes of submanifolds

In this paper, we employ the loop group method to study the construction of minimal Lagrangian surfaces in the complex projective plane for which the domain of the surface is contractible. We present several new classes of minimal Lagrangian surfaces in ℂP2.

We present a geometric interpretation of integrability of geodesic flow by quadratic integrals in terms of the web theory and construct integrable billiards on surfaces admitting such integrals.

The purpose of this article is to present a “Groupoid proof” to the Lefschetz fixed point formula for elliptic complexes whose differentials are of order one. We shall define a “relative version” of tangent groupoid which resemble the appropriate localization phenomena, describe the corresponding groupoid algebra and explain the relation with the Lefschetz fixed point formula.

We investigate the wave equation on Lorentzian conformally flat spaces. In this respect, the problem of the symmetry classification is answered. We find symmetry algebra and construct the Lie subalgebras optimal system. Similarity reductions concerning to subalgebras are classified, and some exact invariant solutions are presented. Finally, for instance, the wave equation and its solution are examined

The paper is devoted to the relation between symplectic structures and variational principles for systems of differential equations. A method for obtaining a global variational principle from a suitable symplectic structure is described. Relation of this result to the inverse problem of the calculus of variations is discussed. It is shown that each variational formulation for a system of evolution

In this paper we give new Gaussian type upper bounds for the Schrödinger heat kernel on complete gradient shrinking Ricci solitons with the scalar curvature bounded above. This result is a little broader than our earlier paper at some cases. The proof uses on a Davies type integral estimate and a local mean value inequality on gradient shrinking Ricci solitons.

Let G be a complex reductive group and XrG denote the G-character variety of the free group of rank r. Using geometric methods, we prove that E(XrSLn)=E(XrPGLn), for any n,r∈N, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety X, settling a conjecture of Lawton–Muñoz in Lawton and Muñoz (2016). The proof involves the stratification by polystable type

We construct two types of Eguchi–Hanson metrics with the negative constant scalar curvature. The type I metrics are Kähler. The type II metrics are ALH whose total energy can be negative.

We construct a class of infinite-dimensional Frobenius manifolds on the spaces of pairs of meromorphic functions with a pole at infinity and a movable pole. Such Frobenius manifolds are shown to be underlying the Whitham hierarchy, which is an extension of the dispersionless Kadomtsev–Petviashvili hierarchy.

We study hypersurfaces of the homogeneous nearly Kähler manifold S3×S3 which have P-isotropic normal vector field. We describe the immersion of such hypersurfaces in S3×S3 and we give one example. We prove that they cannot be either Hopf or minimal hypersurfaces.

Considering matter coupled supersymmetric Chern–Simons theories in three dimensions we extend the Gaiotto–Witten mechanism of supersymmetry enhancement N3=3→N3=4 from the case where the hypermultiplets span a flat HyperKähler manifold to that where they live on a curved one. We derive the precise conditions of this enhancement in terms of generalized Gaiotto–Witten identities to be satisfied by the

The objective, in this paper, is to obtain the curvature properties of (t−z)-type plane wave metric studied by Bondi et al. (1959). For this a general (t−z)-type wave metric is considered and the condition for which it obeys Einstein’s empty spacetime field equations is obtained. It is found that the rank of the Ricci tensor of (t−z)-type plane wave metric is 1 and is of Codazzi type. Also it is proved

We study homogeneous Einstein metrics on indecomposable non-Kähler C-spaces, i.e. even-dimensional torus bundles M=G∕H with rankG>rankH over flag manifolds F=G∕K of a compact simple Lie group G. Based on the theory of painted Dynkin diagrams we present the classification of such spaces. Next we focus on the family Mℓ,m,n≔SU(ℓ+m+n)∕SU(ℓ)×SU(m)×SU(n),ℓ,m,n∈Z+and examine several of its geometric properties

A compatible almost complex structure on a Riemannian manifold can be considered as a smooth map J from the manifold into its twistor space endowed with a natural Riemannian metric induced by the metric of the base manifold or as a self-map J of the unit tangent bundle endowed with the Sasaki metric. The conditions under which the map J is harmonic have been found in Davidov et al. (2018) in the four-dimensional

Recently, the investigation of a CR submanifolds of the nearly Kähler manifold S3×S3 was started. In this paper it is proved that CR submanifolds of the nearly Kähler manifold S3×S3 with umbilical sections must have dimension three and then we obtain some examples of them with distinguished vector fields. Also, we classify minimal submanifolds that have a vector field E4 as an umbilical section. The

Symmetries and differential invariants of viscid flows with viscosity depending on temperature on a space curve are given. Their dependence on thermodynamic states of media is studied, and a classification of thermodynamic states is given.

Let Mn be the Simpson compactification of twisted ideal sheaves IL,Q(1) where Q is a rank 4 quadratic hypersurface in Pn and L is a linear subspace of dimension n−2. This paper calculates the intersection Poincaré polynomial of Mn using Kirwan’s desingularization method. We obtain the intersection Poincaré polynomial of the moduli space for one-dimensional sheaves on del Pezzo surfaces of degree ≥8

A simultaneous description of the dynamics of multiple particles requires a configuration space approach with an external time parameter. This is in stark contrast with the relativistic paradigm, where time is but a coordinate chosen by an observer. Here we show, however, that the two attitudes towards modelling N-particle dynamics can be conciliated within a generally covariant framework. To this

We construct an A∞-structure of the Fukaya category explicitly for any flat symplectic two-torus. The structure constants of the non-transversal A∞-products are obtained as derivatives of those of transversal A∞-products.

In this article, we investigated a nonlinear higher dimensional time fractional Korteweg–de Vries-type (KdV) equation. This considered equation is usually used to describe shallow water waves phenomena in physics. Here we found firstly the symmetry of the higher dimensional time fractional KdV-type equation in the sense of the Riemann–Liouville (RL) fractional derivative with the aid of the fractional