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Frieze groups are discrete subgroups of the full group of isometries of a flat strip. We investigate here the dynamics of specific architected materials generated by acting with a frieze group on a collection of self-coupling seed resonators. We demonstrate that, under unrestricted reconfigurations of the internal structures of the seed resonators, the dynamical matrices of the materials generate the full self-adjoint sector of the stabilized group C*-algebra of the frieze group. As a consequence, in applications where the positions, orientations and internal structures of the seed resonators are adiabatically modified, the spectral bands of the dynamical matrices carry a complete set of topological invariants that are fully accounted by the K-theory of the mentioned algebra. By resolving the generators of the K-theory, we produce the model dynamical matrices that carry the elementary topological charges, which we implement with systems of plate resonators to showcase several applications in spectral engineering. The paper is written in an expository style. [ABSTRACT FROM AUTHOR]

We consider the quantum Gelfand invariants which first appeared in a landmark paper by Reshetikhin et al. [Algebra Anal. 1(1), 178–206 (1989)]. We calculate the eigenvalues of the invariants acting in irreducible highest weight representations of the quantized enveloping algebra for g l n . The calculation is based on Liouville-type formulas relating two families of central elements in the quantum affine algebras of type A. [ABSTRACT FROM AUTHOR]

We study the effects of the branching o s p (1 | 2 n) ⊃ g l (n) on a particular class of simple infinite-dimensional o s p (1 | 2 n) -modules L(p) characterized by a positive integer p. In the first part (Sec. III), we use combinatorial methods, such as Young tableaux and Young subgroups, to construct a new basis for L(p) that respects this branching, and we express the basis elements explicitly in two distinct ways: first, as monomials of negative root vectors of g l (n) acting on certain g l (n) -highest weight vectors in L(p) and then as polynomials in the generators of o s p (1 | 2 n) acting on a o s p (1 | 2 n) -lowest weight vector in L(p). In the second part (Sec. IV), we use extremal projectors and the theory of Mickelsson–Zhelobenko algebras to give new explicit constructions of raising and lowering operators related to the branching o s p (1 | 2 n) ⊃ g l (n). We use the raising operators to give new expressions for the elements of the Gel'fand–Zetlin basis for L(p) as monomials of operators from U (o s p (1 | 2 n)) acting on a o s p (1 | 2 n) -lowest weight vector in L(p). We observe that the Gel'fand–Zetlin basis for L(p) is related to the basis constructed earlier in this paper by a triangular transition matrix. We end this paper (Sec. V) with a detailed example treating the case n = 3. [ABSTRACT FROM AUTHOR]

In this paper, we construct the Heisenberg–Virasoro algebra in the framework of the R (p , q) -deformed quantum algebras. Moreover, the R (p , q) -Heisenberg–Witt n-algebras is also investigated. Furthermore, we generalize the notion of the elliptic Hermitian matrix models. We use the constraints to evaluate the R (p , q) -differential operators of the Virasoro algebra and generalize it to higher order differential operators. Particular cases corresponding to quantum algebras existing in the literature are deduced. [ABSTRACT FROM AUTHOR]

In our previous publications, we have introduced analogs of partial derivatives on the reflection equation algebras, associated with Hecke symmetries. As a consequence, we get quantum partial derivatives on the enveloping algebras U(gl(N)). In the current paper, we consider the particular case N = 2 in detail and discuss the problem of a prolongation of these derivatives onto some central extension of the compact form U(u(2)) of the algebra U(gl(2)). Possible applications of this noncommutative geometry are discussed. [ABSTRACT FROM AUTHOR]

In this paper, a class of non-weight modules over the topological N = 2 super-BMS3 algebra g are completely constructed. Assume that h ̄ = C L 0 ⊕ C P 0 ⊕ C G 0 ⊕ C Q 0 is the Cartan subalgebra of g and h = C L 0 ⊕ C P 0 is a two-dimensional subalgebra of h ̄ . These modules over g are free of rank 2 as modules of the subalgebra h. In fact, these modules are reducible. Moreover, we give a complete classification of free U (h) -modules of rank 2 over g. [ABSTRACT FROM AUTHOR]

In this paper, we define a new type restricted quantum group U ̄ q (s l 2 * ) and determine its Hopf Poincaré-Birkhoff-Witt-deformations U ̄ q (s l 2 * , κ) in which U ̄ q (s l 2 * , 0) = U ̄ q (s l 2 * ) and the classical restricted Drinfeld–Jimbo quantum group U ̄ q (s l 2 ) is included. We show that U ̄ q (s l 2 * ) is a basic Hopf algebra, then uniformly realize U ̄ q (s l 2 * ) and U ̄ q (s l 2 ) via some quotients of (deformed) preprojective algebras corresponding to the Gabriel quiver of U ̄ q (s l 2 * ). Moreover, we obtain a uniform tensor-categorical realization of U ̄ q (s l 2 * ) and U ̄ q (s l 2 ) , which is consistent with the above-mentioned Hopf-algebraic realization. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the affine ageing algebra a g e ̂ (1) , which is a central extension of the loop algebra of the one-spatial ageing algebra a g e (1). Certain Verma-type modules including Verma modules and imaginary Verma modules of a g e ̂ (1) are studied. Particularly, the simplicity of these modules are characterized and their irreducible quotient modules are determined. We also study the restricted modules of a g e ̂ (1) which are also the modules of the affine vertex algebra arising from the one-spatial ageing algebra a g e (1). We present certain constructions of simple restricted a g e ̂ (1) -modules and an explicit such example of simple restricted module via the Whittaker module of a g e ̂ (1) is given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a family of functors denoted as F b , which act on algebraic D-modules and produce modules over N = 2 superconformal algebras. We demonstrate that these functors preserve irreducibility for all values of b, except for explicitly outlined cases. Moreover, we establish the necessary and sufficient conditions for determining the natural isomorphism between two such functors. Our constructed functors recover specific irreducible modules over N = 2 superconformal algebras, including intermediate series and U (h) -free modules. Additionally, we show that our constructed functors yield several new irreducible modules for N = 2 superconformal algebras. [ABSTRACT FROM AUTHOR]

The notion of O -operator is a generalization of the Rota–Baxter operator in the presence of a bimodule over an associative algebra. A compatible O -operator is a pair consisting of two O -operators satisfying a compatibility relation. A compatible O -operator algebra is an algebra together with a bimodule and a compatible O -operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O -operators and compatible O -operator algebras as Maurer–Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible O -operators and compatible O -operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible O -operators. [ABSTRACT FROM AUTHOR]

The Lie superalgebra p s l (2 | 2) is recognized as a quite special algebra. In mathematics, it has the vanishing Killing form and allows for the three-dimensional central extension. In physics, it shows up as the symmetry of the one-dimensional Hubbard model and the asymptotic S-matrix of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. In this paper, we present the Drinfeld realization of the Yangian algebra associated with the centrally extended Lie superalgebra p s l (2 | 2). Furthermore, we show that it possesses Hopf algebra structures, particularly the coproducts. The idea to prove the existence of manifest coproducts is as follows. First, we shall introduce them to Levendorskii's realization, a system of a finite truncation of Drinfeld generators. Second, we show that Levendorskii's realization is isomorphic to the Drinfeld realization by induction. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a series of new invariants for singularities. A new conjecture about the non-existence of negative weight derivations of the new k-th singular locus moduli algebras for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture in some cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce twisted Rota–Baxter operators on Lie algebras as an operator analog of twisted r-matrices. We construct a suitable L∞-algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This allows us to define the cohomology of a twisted Rota–Baxter operator. This cohomology can be seen as the Chevalley–Eilenberg cohomology of a certain Lie algebra with coefficients in a suitable representation. We study deformations of twisted Rota–Baxter operators from cohomological points of view. Some applications are given to Reynolds operators and twisted r-matrices. Next, we introduce a new algebraic structure, called NS-Lie algebras, that is related to twisted Rota–Baxter operators in the same way pre-Lie algebras are related to Rota–Baxter operators. We end this paper by considering twisted generalized complex structures on modules over Lie algebras. [ABSTRACT FROM AUTHOR]

Long time ago, Yang [Phys. Rev. 72, 874 (1947)] proposed a model of noncommutative spacetime that generalized the Snyder model to a curved background. In this paper, we review his proposal and the generalizations that have been suggested during the years. In particular, we discuss the most general algebras that contain as subalgebras both de Sitter and Snyder algebras, preserving Lorentz invariance, and are generated by a two-parameter deformation of the canonical Heisenberg algebra. We also define their realizations on quantum phase space, giving explicit examples, both exact and in terms of a perturbative expansion in deformation parameters. [ABSTRACT FROM AUTHOR]

As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the "top degree component" of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible "top degree component" to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical presymplectic 2-form on the loop space. This construction is a crucial ingredient for the definition of the supersymmetric path integral on the loop space. [ABSTRACT FROM AUTHOR]

In recent years, various nonlinear algebraic structures have been obtained in the context of quantum systems as symmetry algebras, Painlevé transcendent models, and missing label problems. In this paper, we treat all these algebras as instances of the class of quadratic (and higher degree) commutator bracket algebras of Poincaré–Birkhoff–Witt type. We provide a general approach for simplifying the constraints arising from the diamond lemma and apply this in particular to give a comprehensive analysis of the quadratic case. We present new examples of quadratic algebras, which admit a cubic Casimir invariant. The connection with other approaches, such as Gröbner bases, is developed, and we suggest how our explicit and computational techniques can be relevant in other contexts. [ABSTRACT FROM AUTHOR]

In this paper, we define differential graded vertex operator algebras and the algebraic structures on the associated Zhu algebras and C2-algebras. We also introduce the corresponding notions of modules, and investigate the relations between the different module categories. [ABSTRACT FROM AUTHOR]

It was conjectured by Belavin et al. [J. High Energy Phys. 2013(3), 35] that bosonization of a singular vector (in the Neveu–Schwarz sector) of the N = 1 super analog of the Virasoro algebra can be identified with the Uglov symmetric function. In this paper, we prove this conjecture. We also extend this result to the Ramond sector of the N = 1 super-Virasoro algebra. [ABSTRACT FROM AUTHOR]

In this paper, we study non-weight modules over the super-Galilean conformal algebra. We construct and classify U (H) -free modules of rank 1 over the Ramond-type algebra and U (h) -free modules of rank 2 over the Neveu–Schwarz-type algebra, where H is a subalgebra of the Ramond-type algebra and h is the Cartan algebra of the Neveu–Schwarz-type algebra. We find that these modules are reducible and isomorphic. [ABSTRACT FROM AUTHOR]

In a seminal paper, Bacry and Lévy–Leblond classified kinematical algebras, a class of Lie algebras encoding the symmetries of spacetime. Homogeneous spacetimes (infinitesimally, Klein pairs) associated with these possible kinematics can be partitioned into four families—riemannian, lorentzian, galilean, and carrollian—based on the type of invariant metric structure they admit. In this work, we classify possible ambient kinematics—defined as extensions of kinematical algebras by a scalar ideal—as well as their associated Klein pairs. Kinematical Klein pairs arising as quotient space along the extra scalar ideal are said to admit a lift into the corresponding ambient Klein pair. While all non-galilean Klein pairs admit a unique—trivial and torsionfree—higher-dimensional lift, galilean Klein pairs are constructively shown to admit lifts into two distinct families of ambient Klein pairs. The first family includes the bargmann algebra as well as its curved/torsional avatars while the second family is novel and generically allows lifts into torsional ambient spaces. We further comment on the relation between these two families and the maximally symmetric family of leibnizian Klein pairs. [ABSTRACT FROM AUTHOR]

In this paper, we first propose the concept of Rota–Baxter family Ω-associative conformal algebras, then we study the cohomology theory of Rota–Baxter family Ω-associative conformal algebras of any weight and justify it by interpreting the lower degree cohomology groups as formal deformations. [ABSTRACT FROM AUTHOR]

In this paper, the modified KP hierarchy in the Kupershmidt–Kiso version is extended to the super case by Kac–van de Leur construction, that is, using highest weight representations of the even part in the tensor product of the infinite-dimensional Lie superalgebra gl∞|∞ with Grassmann algebra G. First, the super modified KP (SmKP) hierarchy is constructed in terms of superfermionic bilinear equations. Then, the superbosonic form of the SmKP hierarchy is given by super boson–fermion correspondence. With the help of super Hirota bilinear operators, the corresponding super Hirota bilinear equations of the SmKP hierarchy are obtained. Next, the Darboux transformations of this new SmKP hierarchy are expressed in the form of free superfermions and various solutions are derived. Finally, the super bilinear equations in the form of super wave functions are also constructed from the superbosonic ones, which is hoped to be helpful to obtain the corresponding Lax structures. [ABSTRACT FROM AUTHOR]

In this paper, we first prove that the affine Yangian of g l ̂ (1) is isomorphic to the algebra YB 0 g l ̂ (1) whose generators ej, fj, ψj are defined using the Maulik–Okounkov R -matrix. Then, we provide the MacMahon representation of YB g l ̂ (1) which is generated by hj, ej, fj, ψj and find that the representation in the zero twist integrable system is isomorphic to the MacMahon representation. Finally, we discuss a special case in the zero twist integrable system, we obtain one kind of symmetric functions Y λ ( p ⃗) defined on two-dimensional Young diagrams, which are symmetric about the x-axis and y-axis, and the symmetric functions Y λ ( p ⃗) become Jack polynomials and Schur functions in special cases. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notions of quadratic and symplectic structures to the case of 3-(Hom)–ρ-Lie algebras. We describe some properties of them by expressing the related lemmas and theorems. In addition, we introduce the concept of 3-pre-(Hom)–ρ-Lie algebras and define their representation. [ABSTRACT FROM AUTHOR]

Free fermions on Johnson graphs J(n, k) are considered, and the entanglement entropy of sets of neighborhoods is computed. For a subsystem composed of a single neighborhood, an analytical expression is provided by the decomposition in irreducible submodules of the Terwilliger algebra of J(n, k) embedded in two copies of s u (2). For a subsystem composed of multiple neighborhoods, the construction of a block-tridiagonal operator that commutes with the entanglement Hamiltonian is presented, its usefulness in computing the entropy is stressed, and the area law pre-factor is discussed. [ABSTRACT FROM AUTHOR]

In this work, we find Hochschild cohomology groups of the Weyl associative conformal algebra with coefficients in all finite modules. The Weyl conformal algebra is the universal associative conformal envelope of the Virasoro Lie conformal algebra relative to the locality N = 2. In order to obtain this result, we adjust the algebraic discrete Morse theory to the case of differential algebras. [ABSTRACT FROM AUTHOR]

We consider the problem of the R-matrix of the quantum toroidal algebra U q , t ( g l.. 1 ) in the Fock representation. Using the connection between the R-matrix R(u) (u being the spectral parameter) and the theory of Macdonald operators, we obtain explicit formulas for R(u) in the operator and matrix forms. These formulas are expressed in terms of the eigenvalues of a certain Macdonald operator, which completely describe the functional dependence of R(u) on the spectral parameter u. We then consider the geometric R-matrix (obtained from the theory of K-theoretic stable bases on moduli spaces of framed sheaves), which is expected to coincide with R(u) and thus gives another approach to the study of the poles of the R-matrix as a function of u. [ABSTRACT FROM AUTHOR]

An unusual family of generalized coherent states of the Heisenberg algebra is constructed. It consists of two families of functionals on the Schwartz space. Each of them has the key properties of ordinary coherent states: it intertwines the representations of a given algebra, has the completeness property, and minimizes the product of variances in the Heisenberg relation. However, unlike the usual coherent states that are eigenstates for the annihilation operator, the constructed distributions belong to the continuous spectrum of Hermitian generators of the Heisenberg algebra. The inner product of functionals from different families has the main properties of the overlap function of ordinary coherent states: it is continuous in parameters, satisfies the reproducing identity, and has the corresponding geometric meaning. The application of unusual coherent states of the continuous spectrum is demonstrated by an example of solving the spectral problem for an inverted oscillator. [ABSTRACT FROM AUTHOR]

We introduce new combinatorial models called strip bundles for the quantum affine algebra U q ( D 4 (3) ) and give new realizations of the crystal B(∞) and the highest weight crystals B(λ) over U q ( D 4 (3) ) in terms of strip bundles. In these strip bundle realizations, the highest weight crystal B(Λ1) of the natural representation V(Λ1) over the quantum finite algebra Uq(G2) plays a crucial role. [ABSTRACT FROM AUTHOR]

We put forward a definition for spectral triples and algebraic backgrounds based on Jordan coordinate algebras. We also propose natural and gauge-invariant bosonic configuration spaces of fluctuated Dirac operators and compute them for general, almost-associative, Jordan, coordinate algebras. We emphasize that the theory so obtained is not equivalent with usual associative noncommutative geometry, even when the coordinate algebra is the self-adjoint part of a C*-algebra. In particular, in the Jordan case, the gauge fields are always unimodular, thus curing a long-standing problem in noncommutative geometry. [ABSTRACT FROM AUTHOR]

We review the integration of the Kadomtsev–Petviashvili (KP) hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation, an algebra of functions equipped with a derivation that generalizes the gradient operator, an algebra of quaternion-valued functions, a differential Lie algebra, an algebra of polynomials equipped with the Pincherle differential, and a Moyal algebra. In all these cases, we can formulate and solve the Cauchy problem of the KP hierarchy. In addition, in each of these cases, we derive different Zakharov–Shabat (t2, t3)-equations—that is, different Kadomtsev–Petviashvili equations—and we prove the existence of solutions arising from solutions to the corresponding KP hierarchy. [ABSTRACT FROM AUTHOR]

After arguing why the Batalin–Vilkovisky (BV) formalism is expected to find a natural description within the framework of noncommutative geometry, we explain how this relation takes form for gauge theories induced by finite spectral triples. In particular, we demonstrate how the two extension procedures appearing in the BV formalism, that is, the initial extension via the introduction of ghost/anti-ghost fields and the further extension with auxiliary fields, can be described in the language of noncommutative geometry using the notions of the BV spectral triple and total spectral triple, respectively. The construction is presented in detail for all U(2)-gauge theories induced by spectral triples on the algebra M 2 (C). Indications are given on how to extend the results to U(n)-gauge theories for n > 2. [ABSTRACT FROM AUTHOR]

Using the Zhu algebra for a certain category of C -graded vertex algebras V, we prove that if V is finitely Ω-generated and satisfies suitable grading conditions, then V is rational, i.e., it has semi-simple representation theory, with a one-dimensional level zero Zhu algebra. Here, Ω denotes the vectors in V that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element ωμ parameterized by μ ∈ C and prove that for certain non-integer values of μ, these vertex algebras, which are non-integer graded, are rational, with a one-dimensional level zero Zhu algebra. In addition, we generalize this result to appropriate C -graded Weyl vertex algebras of arbitrary ranks. [ABSTRACT FROM AUTHOR]

Weighted Rota–Baxter operators on associative algebras are closely related to modified Yang–Baxter equations, splitting of algebras, and weighted infinitesimal bialgebras and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by λ-weighted relative Rota–Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal, and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation, which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota–Baxter operators in the Lie case and find a connection with the case of associative algebras. [ABSTRACT FROM AUTHOR]

Within the framework of the algebra of canonical commutation relations in position space, a long range order between particles in bounded regions is established in states with a sufficiently large particle number. It occurs whenever homogeneous proper (infinite) condensates form locally in the states in the limit of infinite densities. The condensates are described by eigenstates of the momentum operator, covering also those cases where they are streaming with a constant velocity. The arguments given are model independent and lead to a new criterion for the occurrence of condensates. It makes use of a novel approach to the identification of condensates, based on a characterization of regular and singular wave functions. [ABSTRACT FROM AUTHOR]

We aim to explore if, inside a quantum vertex algebra, we can find the right notion of a quantum conformal algebra. [ABSTRACT FROM AUTHOR]

We interpret the elements of the exceptional Lie algebra e 8 (− 24) as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra s u (3) ⊕ s u (2) ⊕ u (1) , and the Lorentz Lie algebra s o (3 , 1). Our construction relies on identifying a complex structure on spinors and then working in the enveloping algebra. The resulting model naturally contains Grand Unified Theories based on SO(10) (Georgi), SU(5) (Georgi–Glashow), and SU(4) ×SU(2) ×SU(2) (Pati–Salam). We then briefly speculate on the role of the remaining elements of e 8 and propose a mechanism leading to exactly three generations of particles. [ABSTRACT FROM AUTHOR]

We introduce and study a large class of algebras (possibly nonassociative) with group-coalgebraic structures, Hopf nonassociative group-coalgebras, which provide a unifying framework for classical Hopf algebras, Turaev's Hopf group-coalgebras, and Hopf quasigroups. Then, we introduce and discuss the notion of a quasitriangular Hopf π-quasialgebra and show that the category Qrep(H)R of quasirepresentations over any quasitriangular Hopf π-quasialgebra (H, R) is a braided π-category. As an application of our theory, we give a new solution to the generalized quantum Yang–Baxter equation. [ABSTRACT FROM AUTHOR]

The polynomials discovered by Chebyshev and subsequently related to spin transition probabilities by Meckler [Meckler, Phys. Rev. 111, 1447 (1958)] and Mermin and Schwarz [Found. Phys. 12, 101 (1982)] are studied, and their application to phase space representations of spin states and operators is examined. In particular, a formula relating the end-point value of the polynomials to scale factors relating different phase space representations of spherical harmonic operators is found. This formula is applied to illustrative calculations of Wigner functions for a single spin and the singlet state of a pair of spins. [ABSTRACT FROM AUTHOR]

We introduce several definitions within the framework of vertex and conformal algebras that are analogous to some important concepts of the classical Lie theory. Most importantly, we define formal vertex laws, which correspond to the notion of formal group laws. We prove suitable vertex/conformal versions of a number of classical results, such as the Milnor–Moore theorem, Cartier duality, and the equivalence between formal group laws and Lie algebras. [ABSTRACT FROM AUTHOR]

Within the setting of infinite-dimensional self-dual CAR C* algebras describing fermions in the Z d lattice, we depart from the well-known Araki–Evans σ ( P 1 , P 2 ) Z 2 index for quasi-free fermion states and rewrite it in terms of states rather than in terms of basis projections. Furthermore, we reformulate results that relate equivalences of Fock representations to the index parity into results that relate equivalences of Gel'fand–Naimark–Segal representations and the associated index parity. [ABSTRACT FROM AUTHOR]

The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second-order Heun operators on quadratic grids with no diagonal terms are determined. These special or S-Heun operators lead to the rational degeneration of the Sklyanin algebra; they also entail the contiguity and structure operators of the Wilson polynomials. The finite-dimensional restriction yields a representation that acts on the para-Racah polynomials. [ABSTRACT FROM AUTHOR]

A relative Rota–Baxter algebra is a triple (A, M, T) consisting of an algebra A, an A-bimodule M, and a relative Rota–Baxter operator T. Using Voronov's derived bracket and a recent work of Lazarev, Sheng, and Tang, we construct an L∞[1]-algebra whose Maurer–Cartan elements are precisely relative Rota–Baxter algebras. By a standard twisting, we define a new L∞[1]-algebra that controls Maurer–Cartan deformations of a relative Rota–Baxter algebra (A, M, T). We introduce the cohomology of a relative Rota–Baxter algebra (A, M, T) and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of triangular skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota–Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras. [ABSTRACT FROM AUTHOR]

We construct a (1 + 1)d topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup H 3 fusion category. This TFT has six vacua, and each of the three non-invertible simple TDLs hosts three defect operators, giving rise to a total of 15 point-like operators. The TFT data, including the three-point coefficients and lasso diagrams, are determined by solving all the sphere four-point crossing equations and torus one-point modular invariance equations. We further verify that the Cardy states furnish a non-negative integer matrix representation under TDL fusion. While many of the constraints we derive are not limited to this particular TFT with six vacua, we leave open the construction of TFTs with two or four vacua. Finally, TFTs realizing the Haagerup H 1 and H 2 fusion categories can be obtained by gauging algebra objects. This article makes a modest offering in our pursuit of exotica and the quest for their eventual conformity. [ABSTRACT FROM AUTHOR]

We discuss an embedding of su(n) rank two antisymmetric supercharges in the su(2, 2|dn) superalgebra, where dn = n(n − 1)/2. We describe an algorithm to construct the explicit form of the generators of the superalgebra. [ABSTRACT FROM AUTHOR]

The algebras of the integrals of motion of the Hamilton–Jacobi and Klein–Gordon–Fock equations for a charged test particle moving in an external electromagnetic field in a spacetime manifold are found. The manifold admits four-parameter groups of motions that act nontransitively on the spacetime. All admissible electromagnetic fields for which such algebras exist are found. In the case of an arbitrary n-dimensional Riemannian space on which the groups of motions act, it is proved that the admissible field does not deform the algebra of symmetry operators of the free Hamilton–Jacobi and Klein–Gordon–Fock equations. In addition, the system of differential equations, which must be satisfied by the potentials of the admissible electromagnetic field, has been investigated for compatibility. [ABSTRACT FROM AUTHOR]

We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, "generalized"), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called "super-adiabatic"). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article. [ABSTRACT FROM AUTHOR]

We associate a deformation of the Heisenberg algebra with the suitably normalized Yang R-matrix, and we investigate its properties. Moreover, we construct new examples of quantum vertex algebras, which possess the same representation theory as the aforementioned deformed Heisenberg algebra. [ABSTRACT FROM AUTHOR]

We calculate the second Hochschild cohomology group with scalar coefficients for the universal enveloping associative conformal algebra of the Virasoro conformal algebra relative to the locality N = 3 on the generator. For the associative conformal envelope corresponding to smaller locality N = 2, i.e., for the Weyl conformal algebra, we show that all Hochschild cohomologies with scalar coefficients are vanishing. [ABSTRACT FROM AUTHOR]

We study the quantization of the corner symmetry algebra of 3D gravity, that is, the algebra of observables associated with 1D spatial boundaries. In the continuum field theory, at the classical level, this symmetry algebra is given by the central extension of the Poincaré loop algebra. At the quantum level, we construct a discrete current algebra based on a quantum symmetry group given by the Drinfeld double D S U (2). Those discrete currents depend on an integer N, a discreteness parameter, understood as the number of quanta of geometry on the 1D boundary: low N is the deep quantum regime, while large N should lead back to a continuum picture. We show that this algebra satisfies two fundamental properties. First, it is compatible with the quantum space-time picture given by the Ponzano–Regge state-sum model, which provides discrete path integral amplitudes for 3D quantum gravity. The integer N then counts the flux lines attached to the boundary. Second, we analyze the refinement, coarse-graining, and fusion processes as N changes, and we show that the N → ∞ limit is a classical limit where we recover the Poincaré current algebra. Identifying such a discrete current algebra on quantum boundaries is an important step toward understanding how conformal field theories arise on spatial boundaries in quantized space-times such as in loop quantum gravity. [ABSTRACT FROM AUTHOR]