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In this paper, we present an extended dissipaton equation of motion for studying the dynamics of electronic impurity systems. Compared with the original theoretical formalism, the quadratic couplings are introduced into the Hamiltonian accounting for the interaction between the impurity and its surrounding environment. By exploiting the quadratic fermionic dissipaton algebra, the proposed extended dissipaton equation of motion offers a powerful tool for studying the dynamical behaviors of electronic impurity systems, particularly in situations where nonequilibrium and strongly correlated effects play significant roles. Numerical demonstrations are carried out to investigate the temperature dependence of the Kondo resonance in the Kondo impurity model. [ABSTRACT FROM AUTHOR]

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]