Search

This paper studies the abelian subalgebras and ideals of maximal dimension of Poisson algebras P of dimension n. We introduce the invariants α and β for Poisson algebras, which correspond to the dimension of an abelian subalgebra and ideal of maximal dimension, respectively. We prove that these invariants coincide if α (P) = n − 1. We characterize the Poisson algebras with α (P) = n − 2 over arbitrary fields. In particular, we characterize Lie algebras L with α (L) = n − 2. We also show that α (P) = n − 2 for nilpotent Poisson algebras implies β (P) = n − 2. Finally, we study these invariants for various distinguished Poisson algebras, providing us with several examples and counterexamples. [ABSTRACT FROM AUTHOR]

The notions of the Novikov deformation of a commutative associative algebra and the corresponding classical limit are introduced. We show such a classical limit belongs to a subclass of transposed Poisson algebras, and hence the Novikov deformation is defined to be the quantization of the corresponding transposed Poisson algebra. As a direct consequence, we revisit the relationship between transposed Poisson algebras and Novikov–Poisson algebras due to the fact that there is a natural Novikov deformation of the commutative associative algebra in a Novikov–Poisson algebra. Hence all transposed Poisson algebras of Novikov–Poisson type, including unital transposed Poisson algebras, can be quantized. Finally, we classify the quantizations of 2-dimensional complex transposed Poisson algebras in which the Lie brackets are non-abelian up to equivalence. [ABSTRACT FROM AUTHOR]

This study introduces the notions of BiHom-Poisson conformal algebra, BiHom-pre-Poisson conformal algebra, and their related structures. We show that many new BiHom-Poisson conformal algebras can be constructed from a BiHom-Poisson conformal algebra. In particular, the direct product of two BiHom-Poisson conformal algebras is also a BiHom-Poisson conformal algebra. We further describe the conformal bimodule and representation theory of the BiHom-Poisson conformal algebra. In addition, we define BiHom-pre-Poisson conformal algebra as the combination of BiHom-pre-Lie conformal algebra and BiHom-dendriform conformal algebra under some compatibility conditions. We further demonstrate a way to construct BiHom-Poisson conformal algebra from BiHom-pre-Poisson conformal algebra and provide the representation theory for BiHom-pre-Poisson conformal algebra. Finally, a detailed description of O -operators and Rota–Baxter operators on BiHom-Poisson conformal algebra is provided. [ABSTRACT FROM AUTHOR]

In this paper, we present some generalizations of Jacobi–Jordan algebras. More concretely, we will focus on noncommutative Jacobi–Jordan algebras, Malcev–Jordan algebras, and general Jacobi–Jordan algebras. We adapt a method, used to classify Poisson algebras, in order to classify all general Jacobi–Jordan algebras up to dimension 4, and, in particular, all noncommutative Jacobi–Jordan algebras up to dimension 4. We present the classification of Malcev–Jordan algebras up to dimension 5. As the class of Jacobi–Jordan algebras (commutative algebras that satisfy the Jacobi identity), we find that Malcev–Jordan algebras are Jordan algebras but not necessarily nilpotent. However, we show that the classification of nilpotent Malcev–Jordan algebras is sufficient to obtain the classification of the whole class. [ABSTRACT FROM AUTHOR]

Hilbert bimodules are morphisms between C*-algebraic models of quantum systems, while symplectic dual pairs are morphisms between Poisson geometric models of classical systems. Both of these morphisms preserve representation-theoretic structures of the relevant types of models. Previously, it has been shown that one can functorially associate certain symplectic dual pairs to Hilbert bimodules through strict deformation quantization. We show that, in the inverse direction, strict deformation quantization also allows one to functorially take the classical limit of a Hilbert bimodule to reconstruct a symplectic dual pair. [ABSTRACT FROM AUTHOR]

We study a family of completely integrable systems attached to Takiff algebras g N , extending Toda systems for split simple Lie algebras g. With respect to Darboux coordinates on coadjoint orbits O , the potentials of the hamiltonians are products of polynomial and exponential functions. Explicit general solutions for their equations of motion are obtained using differential operators called jet transformations. The classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, quantizing the Poisson algebra of functions on O. These results are illustrated with concrete examples, including a 3-body problem based on sl (2) and an extension of soliton solutions for A ∞ to associated Takiff algebras. [ABSTRACT FROM AUTHOR]

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang–Baxter equation and -operators are studied. [ABSTRACT FROM AUTHOR]

It is well-known that a formal deformation of a commutative algebra A leads to a Poisson bracket on A and that the classical limit of a derivation on the deformation leads to a derivation on A , which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra A . The deformation leads in this case to a Poisson algebra structure on Π (A) : = Z (A) × (A / Z (A)) and to the structure of a Π (A) -Poisson module on A . The limiting derivations are then still derivations of A , but with the Hamiltonian belong to Π (A) , rather than to A . We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra. [ABSTRACT FROM AUTHOR]

The n-th Schrödinger algebra 픰픠픥n := 픰픩2 ⋉ 픥n is the semidirect product of the Lie algebra 픰픩2 with the n-th Heisenberg Lie algebra 픥n. In this paper, we will show that 픰픠픥n has neither nontrivial 1 2-derivations nor nontrivial transposed Poisson algebra structures, and doesn’t have nonzero 1 2-biderivations. In addition, all δ-derivations and δ-biderivations on 픰픠픥n are all described. [ABSTRACT FROM AUTHOR]

Similar to the modular vector fields in Poisson geometry, modular derivations are defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson module with the modular derivation, the Poisson cochain complex with values in any Poisson module is proved to be isomorphic to the Poisson chain complex with values in the corresponding twisted Poisson module. Then a version of twisted Poincaré duality is proved between the Poisson homologies and cohomologies. Furthermore, a notion of pseudo-unimodular Poisson structure is defined. It is proved that the Poisson cohomology as a Gerstenhaber algebra admits a Batalin-Vilkovisky operator inherited from some one of its Poisson cochain complex if and only if the Poisson structure is pseudo-unimodular. This generalizes the geometric version due to P. Xu. The modular derivation and Batalin-Vilkovisky operator are also described by using the dual basis of the Kähler differential module. [ABSTRACT FROM AUTHOR]

Jacobi algebras, as the algebraic counterparts of Jacobi manifolds, are exactly the unital relative Poisson algebras. The direct approach of constructing Frobenius Jacobi algebras in terms of Manin triples is not available due to the existence of the units, and hence alternatively we replace it by studying Manin triples of relative Poisson algebras. Such structures are equivalent to certain bialgebra structures, namely, relative Poisson bialgebras. The study of coboundary cases leads to the introduction of the relative Poisson Yang–Baxter equation (RPYBE). Antisymmetric solutions of the RPYBE give coboundary relative Poisson bialgebras. The notions of O-operators of relative Poisson algebras and relative pre-Poisson algebras are introduced to give antisymmetric solutions of the RPYBE. A direct application is that relative Poisson bialgebras can be used to construct Frobenius Jacobi algebras, and in particular, there is a construction of Frobenius Jacobi algebras from relative pre-Poisson algebras. [ABSTRACT FROM AUTHOR]

Let X be a finite connected poset, K a field of characteristic zero and I (X , K) the incidence algebra of X over K seen as a Lie algebra under the commutator product. In the first part of the paper we show that any 1 2 -derivation of I (X , K) decomposes into the sum of a central-valued 1 2 -derivation, an inner 1 2 -derivation and a 1 2 -derivation associated with a map σ : X < 2 → K that is constant on chains and cycles in X. In the second part of the paper we use this result to prove that any transposed Poisson structure on I (X , K) is the sum of a structure of Poisson type, a mutational structure and a structure determined by λ : X e 2 → K , where X e 2 is the set of (x , y) ∈ X 2 such that x < y is a maximal chain not contained in a cycle. [ABSTRACT FROM AUTHOR]

Contemporary quantum computers open up novel possibilities for tackling intricate problems, encompassing quantum system modeling and solving partial differential equations (PDEs). This paper explores the optimization of quantum algorithms aimed at resolving PDEs, presenting a significant challenge within the realm of computational science. The work delves into the application of the Variational Quantum Eigensolver (VQE) for addressing equations such as Poisson’s equation. It employs a Hamiltonian constructed using a modified Feynman-Kitaev formalism for a VQE, which represents a quantum system and encapsulates information pertaining to the classical system. By optimizing the parameters of the quantum circuit that implements this Hamiltonian, it becomes feasible to achieve minimization, which corresponds to the solution of the original classical system. The modification optimizes quantum circuits by minimizing the cost function associated with the VQE. The efficacy of this approach is demonstrated through the illustrative example of solving the Poisson equation. The prospects for its application to the integration of more generalized PDEs are discussed in detail. This study provides an in-depth analysis of the potential advantages of quantum algorithms in the domain of numerical solutions for the Poisson equation and emphasizes the significance of continued research in this direction. By leveraging quantum computing capabilities, the development of more efficient methodologies for solving these equations is possible, which could significantly transform current computational practices. The findings of this work underscore not only the practical advantages but also the transformative potential of quantum computing in addressing complex PDEs. Moreover, the results obtained highlight the critical need for ongoing research to refine these techniques and extend their applicability to a broader class of PDEs, ultimately paving the way for advancements in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

By using the classical Lie algebra, the stationary zero curvature equation, and the Lenard recursion equations, we obtain the non-isospectral TD hierarchy. Two kinds of expanding higher-dimensional Lie algebras are presented by extending the classical Lie algebra. By solving the expanded non-isospectral zero curvature equations, the multi-component non-isospectral TD hierarchies are derived. The Hamiltonian structure for one of them is obtained by using the trace identity. [ABSTRACT FROM AUTHOR]

The loop group |$G((z^{-1}))$| of a simple complex Lie group |$G$| has a natural Poisson structure. We introduce a natural family of Poisson commutative subalgebras |$\overline{{\textbf{B}}}(C) \subset{{\mathcal{O}}}(G((z^{-1}))$| depending on the parameter |$C\in G$| called classical universal Bethe subalgebras. To every antidominant cocharacter |$\mu $| of the maximal torus |$T \subset G$|⁠ , one can associate the closed Poisson subspace |${{\mathcal{W}}}_{\mu }$| of |$G((z^{-1}))$| (the Poisson algebra |${{\mathcal{O}}}({{\mathcal{W}}}_{\mu })$| is the classical limit of so-called shifted Yangian |$Y_{\mu }(\mathfrak{g})$| defined in [ 1 , Appendix B]). We consider the images of |$\overline{{\textbf{B}}}(C)$| in |${{\mathcal{O}}}({{\mathcal{W}}}_{\mu })$|⁠ , denoted by |$\overline{B}_{\mu }(C)$|⁠ , that should be considered as classical versions of (not yet defined in general) Bethe subalgebras in shifted Yangians. For regular |$C$| centralizing |$\mu $|⁠ , we compute the Poincaré series of these subalgebras. For |$\mathfrak{g}=\mathfrak{g}\mathfrak{l}_{n}$|⁠ , we define the natural quantization |${\textbf{Y}}^{\textrm{rtt}}(\mathfrak{g}\mathfrak{l}_{n})$| of |${{\mathcal{O}}}(\operatorname{Mat}_{n}((z^{-1})))$| and universal Bethe subalgebras |${\textbf{B}}(C) \subset{\textbf{Y}}^{\textrm{rtt}}(\mathfrak{g}\mathfrak{l}_{n})$|⁠. Using the RTT realization of |$Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$| (invented by Frassek, Pestun, and Tsymbaliuk), we obtain the natural surjections |${\textbf{Y}}^{\textrm{rtt}}(\mathfrak{g}\mathfrak{l}_{n}) \twoheadrightarrow Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$|⁠ , which quantize the embedding |${{\mathcal{W}}}_{\mu } \subset \operatorname{Mat}_{n}((z^{-1}))$|⁠. Taking the images of |${\textbf{B}}(C)$| in |$Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$|⁠ , we recover Bethe subalgebras |$B_{\mu }(C) \subset Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$| proposed by Frassek, Pestun, and Tsymbaliuk. [ABSTRACT FROM AUTHOR]

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of their bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland's representation-theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all known cluster algebras arising from representation theory or higher Teichmüller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds. [ABSTRACT FROM AUTHOR]

We prove that a transposed Poisson algebra is simple if and only if its associated Lie bracket is simple. Consequently, any simple finite-dimensional transposed Poisson algebra over an algebraically closed field of characteristic zero is trivial. Similar results are obtained for transposed Poisson superalgebras. An example of a non-trivial simple finite-dimensional transposed Poisson algebra is constructed by studying the transposed Poisson structures on the modular Witt algebra. Furthermore, we show that the Kantor double of a transposed Poisson algebra is a Jordan superalgebra, that is, we prove that transposed Poisson algebras are Jordan brackets. Additionally, a simplicity criterion for the Kantor double of a transposed Poisson algebra is obtained. [ABSTRACT FROM AUTHOR]

To present a survey on known results from the theory of transposed Poisson algebras, as well as to establish new results on this subject, are the main aims of the present paper. Furthermore, a list of open questions for future research is given. [ABSTRACT FROM AUTHOR]

The notion of a transposed Poisson n -Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson n -Lie algebra with a derivation gives rise to a transposed Poisson (n + 1) -Lie algebra. In this paper, we focus on transposed Poisson n -Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of (n + 1) -Lie algebras from transposed Poisson n -Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases. The notion of a transposed Poisson -Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson -Lie algebra with a derivation gives rise to a transposed Poisson -Lie algebra. In this paper, we focus on transposed Poisson -Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of -Lie algebras from transposed Poisson -Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases. [ABSTRACT FROM AUTHOR]

In his pioneering paper [Phys. Rev. E 7 , 2405 (1973)], Nambu proposed the idea of multiple Hamiltonian systems. The explicit example examined there is equivalent to the |$\mathfrak {so}(3)$| Lie–Poisson system, which represents noncanonical Hamiltonian dynamics with a Casimir; the Casimir corresponds to the second Hamiltonian of Nambu's formulation. The vortex dynamics of an ideal fluid, while it is infinite dimensional, has a similar structure, in which the Casimir is the helicity. These noncanonical Poisson algebras are derived by the reduction, i.e. restricting the phase space to some submanifold embedded in the canonical phase space. We may reverse the reduction to canonicalize some Nambu dynamics, i.e. view the Nambu dynamics as the subalgebra of a larger canonical Poisson algebra. Then, we can invoke the standard corresponding principle for quantizing the canonicalized system. The inverse of the reduction, i.e. representing the noncanonical variables by some canonical variables may be called "Clebsch parameterization" following the fluid mechanical example. [ABSTRACT FROM AUTHOR]

Symmetric space sine-Gordon theories are two-dimensional massive integrable field theories, generalising the sine-Gordon and complex sine-Gordon theories. To study their integrability properties on the real line, it is necessary to introduce a subtracted monodromy matrix. Moreover, since the theories are not ultralocal, a regularisation is required to compute the Poisson algebra for the subtracted monodromy. In this article, we regularise and compute this Poisson algebra for certain configurations, and show that it can both satisfy the Jacobi identity and imply the existence of an infinite number of conserved quantities in involution. [ABSTRACT FROM AUTHOR]

To every h + N -graded module M over an N -graded conformal vertex algebra V , we associate an increasing filtration (G p M) p ∈ Z , which is compatible with the filtrations introduced by Haisheng Li. The associated graded vector space gr G (M) is naturally a module over the vertex Poisson algebra gr G (V). We study gr G (M) for the three irreducible modules over the Ising model Vir 3 , 4 , namely Vir 3 , 4 = L (1 / 2 , 0) , L (1 / 2 , 1 / 2) and L (1 / 2 , 1 / 16). We obtain an explicit PBW basis of each of these modules and a formula for their refined characters, which are related to Nahm sums for the matrix ( 8 3 3 2 ). [ABSTRACT FROM AUTHOR]

This article focuses on a recently developed formulation based on the noncommutative branch‐cut cosmology, the Wheeler‐DeWitt (WdW) equation, the Hořava–Lifshitz quantum gravity, chaotic and the coupling of the corresponding Lagrangian approach with the inflaton scalar field. Assuming a mini‐superspace of variables obeying the noncommutative Poisson algebra, we examine the impact of the inflaton scalar field on the evolutionary dynamics of the branch‐cut Universe scale factor, characterized by the dimensionless helix‐like function ln−1[β(t)]$$ {\ln}^{-1}\left[\beta (t)\right] $$. This scale factor characterizes a Riemannian foliated spacetime that topologically overcomes the primordial singularities. We take the Hořava–Lifshitz action modeled by branch‐cut quantum gravity as our starting point, which depends on the scalar curvature of the branched Universe and its derivatives and which preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner formalism. We then investigate the sensitivity of the scale factor of the branch‐cut Universe's dynamics. [ABSTRACT FROM AUTHOR]

This article focuses on the implications of a noncommutative formulation of branch‐cut quantum gravity. Based on a mini‐superspace structure that obeys the noncommutative Poisson algebra, combined with the Wheeler–DeWitt equation and Hořava–Lifshitz quantum gravity, we explore the impact of a scalar field of the inflaton‐type in the evolution of the Universe's wave function. Taking as a starting point the Hořava–Lifshitz action, which depends on the scalar curvature of the branched Universe and its derivatives, the corresponding wave equations are derived and solved. The noncommutative quantum gravity approach adopted preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner Formalism. In this work we delve deeper into a mini‐superspace of noncommutative variables, incorporating scalar inflaton fields and exploring inflationary models, particularly chaotic and nonchaotic scenarios. We obtained solutions to the wave equations without resorting to numerical approximations. The results indicate that the noncommutative algebraic space captures low and high spacetime scales, driving the exponential acceleration of the Universe. [ABSTRACT FROM AUTHOR]

We introduce the universal algebra of two Poisson algebras P and Q as a commutative algebra A : = P (P , Q) satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional Poisson algebra P and several of its applications are highlighted. For any Poisson P -module U , we construct a functor U ⊗ − : A M → Q P M from the category of A -modules to the category of Poisson Q -modules which has a left adjoint whenever U is finite dimensional. Similarly, if V is an A -module, then there exists another functor − ⊗ V : P P M → Q P M connecting the categories of Poisson representations of P and Q and the latter functor also admits a left adjoint if V is finite dimensional. If P is n -dimensional, then P (P) : = P (P , P) is the initial object in the category of all commutative bialgebras coacting on P. As an algebra, P (P) can be described as the quotient of the polynomial algebra k [ X i j | i , j = 1 , ⋯ , n ] through an ideal generated by 2 n 3 non-homogeneous polynomials of degree ≤2. Two applications are provided. The first one describes the automorphisms group Aut Poiss (P) as the group of all invertible group-like elements of the finite dual P (P) o. Secondly, we show that for an abelian group G , all G -gradings on P can be explicitly described and classified in terms of the universal coacting bialgebra P (P). [ABSTRACT FROM AUTHOR]

In this paper, we propose the notion of a transposed Poisson superalgebra.We prove that a transposed Poisson superalgebra can be constructed by means of a commutative associative superalgebra and an even degree derivation of this algebra. Making use of this, we construct two examples of the transposed Poisson superalgebra. One of them is the graded differential algebra of differential forms on a smooth finite dimensional manifold, where we use the Lie derivative as an even degree derivation. The second example is the commutative superalgebra of basic fields of the quantum Yang--Mills theory, where we use the BRST-supersymmetry as an even degree derivation to define a graded Lie bracket. We show that a transposed Poisson superalgebra has six identities that play an important role in the study of the structure of this algebra. [ABSTRACT FROM AUTHOR]

Given a simple transient random walk (Sn)n=0 in Z and a stationary sequence of real random variables ((s))sZ, we investigate the extremes of the sequence ((Sn))n=0. Under suitable conditions, we make explicit the extremal index and show that the point process of exceedances converges to a compound Poisson point process. We give two examples for which the cluster size distribution can be made explicit. [ABSTRACT FROM AUTHOR]

We study a d-dimensional branching Brownian motion (BBM) among Poissonian obstacles, where a random trap field in Rd is created via a Poisson point process. In the soft obstacle model, the trap field consists of a positive potential which is formed as a sum of a compactly supported bounded function translated at the atoms of the Poisson point process. Particles branch at the normal rate outside the trap field; and when inside the trap field, on top of complete suppression of branching, particles are killed at a rate given by the value of the potential. Under soft killing, the probability that the entire BBM goes extinct due to killing is positive in almost every environment. Conditional on ultimate survival of the process, we prove a law of large numbers for the total mass of BBM among soft Poissonian obstacles. Our result is quenched, that is, it holds in almost every environment with respect to the Poisson point process. [ABSTRACT FROM AUTHOR]

We describe transposed Poisson structures on the upper triangular matrix Lie algebra Tn(F), n > 1, over a field F of characteristic zero. We prove that, for n > 2, any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for n = 2, there is one more class of transposed Poisson structures on Tn(F). We also show that, up to isomorphism, the full matrix Lie algebra Mn(F) admits only one non-trivial transposed Poisson structure, and it is of Poisson type. [ABSTRACT FROM AUTHOR]

We describe a N = 2 supersymmetric Poisson vertex algebra structure of N = 1 (resp. N = 0 ) classical W-algebra associated with s l (n + 1 | n) and the odd (resp. even) principal nilpotent element. This N = 2 supersymmetric structure is connected to the principal s l (2 | 1) -embedding in s l (n + 1 | n) superalgebras, which are the only basic Lie superalgebras that admit such a principal embedding. [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]

In a recent paper, one of the authors proposed a construction of associative algebras which share a number of properties of the Yangians of series A but are more massive. We show that this construction admits a generalization which reveals a direct connection with a large family of double Poisson brackets on free associative algebras, which was described by Pichereau and Van de Weyer (in 2008). [ABSTRACT FROM AUTHOR]

The Kac-Ornstein-Uhlenbeck processes with reset are studied in detail, including the joint distribution of the process and the corresponding counting Poisson process. The Laplace transform of the first passage time is also presented. [ABSTRACT FROM AUTHOR]

This paper surveys the theory of functional identities and its applications. No prior knowledge of the theory is required to follow the paper. [ABSTRACT FROM AUTHOR]

Logarithmic vertex algebras were introduced in our previous paper, motivated by logarithmic conformal field theory (Bakalov and Villarreal in Logarithmic vertex algebras, 2022). Non-local Poisson vertex algebras were introduced by De Sole and Kac, motivated by the theory of integrable systems (De Sole and Kac in Jpn J Math 8:233–347, 2013). We prove that the associated graded vector space of any filtered logarithmic vertex algebra has an induced structure of a non-local Poisson vertex algebra. We use this relation to obtain new examples of both logarithmic vertex algebras and non-local Poisson vertex algebras. [ABSTRACT FROM AUTHOR]

We show that free algebras of the variety of algebras generated by the Witt algebra W n , the left-symmetric Witt algebra L n , and the symplectic Poisson algebra P n can be described as subalgebras of differential polynomial algebras with respect to appropriately defined products. Using these representations, we prove that W n , L n , P n , and the free algebras of the varieties of algebras generated by these algebras are equationally Noetherian. [ABSTRACT FROM AUTHOR]

We introduce a dual notion of the Poisson algebra, called the transposed Poisson algebra, by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. The transposed Poisson algebra shares common properties of the Poisson algebra and arises naturally from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. Consequently, the classic construction of a Poisson algebra from a commutative algebra with two commuting derivations similarly applies to a transposed Poisson algebra. More broadly, the transposed Poisson algebra captures the algebraic structures when the commutator is taken in pre-Lie Poisson algebras and two other Poisson type algebras. Furthermore, the transposed Poisson algebra improves two processes that produce 3-Lie algebras from Poisson algebras with a strongness condition. [ABSTRACT FROM AUTHOR]

In the group algebra of the symmetric group , we consider the subalgebra consisting of all functions invariant with respect to left and right shifts by elements of the Young subgroup . We discuss structure constants of the algebra and construct an algebra with continuous parameters extrapolating algebras , this can also be rewritten as an asymptotic algebra as (for fixed ). We show that there is a natural map from the Lie algebra of the group of pure braids to (and therefore this Lie algebra acts in spaces of multiplicities of the quasiregular representation of the group in functions on ). [ABSTRACT FROM AUTHOR]

We analyse the Noether charges for scalar and Maxwell fields on light cones on a de Sitter, Minkowski, and anti-de Sitter backgrounds. Somewhat surprisingly, under natural asymptotic conditions all charges for the Maxwell fields on both the de Sitter and anti-de Sitter backgrounds are finite. On the other hand, one needs to renormalise the charges for the conformally-covariant scalar field when the cosmological constant does not vanish. In both cases well-defined renormalised charges, with well-defined fluxes, are obtained. Again surprisingly, a Hamiltonian analysis of a suitably rescaled scalar field leads to finite charges, without the need to renormalise. Last but not least, we indicate natural phase spaces where the Poisson algebra of charges is well defined. [ABSTRACT FROM AUTHOR]

We describe transposed Poisson structures on generalized Witt algebras W (A , V , ⟨ · , · ⟩) and Block Lie algebras L(A, g, f) over a field F of characteristic zero, where ⟨ · , · ⟩ and f are non-degenerate. More specifically, if dim (V) > 1 , then all the transposed Poisson algebra structures on W (A , V , ⟨ · , · ⟩) are trivial; and if dim (V) = 1 , then such structures are, up to isomorphism, mutations of the group algebra structure on FA. The transposed Poisson algebra structures on L(A, g, f) are in a one-to-one correspondence with commutative and associative multiplications defined on a complement of the square of L(A, g, f) with values in the center of L(A, g, f). In particular, all of them are usual Poisson structures on L(A, g, f). This generalizes earlier results about transposed Poisson structures on Block Lie algebras B (q) . [ABSTRACT FROM AUTHOR]

The first historical encounter with Poisson-type algebras is with Hamiltonian mechanics. With the abstraction of many notions in Physics, Hamiltonian systems were geometrized into manifolds that model the set of all possible configurations of the system, and the cotangent bundle of this manifold describes its phase space, which is endowed with a Poisson structure. Poisson brackets led to other algebraic structures, and the notion of Poisson-type algebra arose, including transposed Poisson algebras, Novikov-Poisson algebras, or commutative pre-Lie algebras, for example. These types of algebras have long gained popularity in the scientific world and are not only of their own interest to study, but are also an important tool for researching other mathematical and physical objects. [ABSTRACT FROM AUTHOR]

In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and give some constructions of them. Then we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations. [ABSTRACT FROM AUTHOR]

We establish the structure of the dual Lie coalgebra for a Lie algebra of the symplectic Poisson bracket (Jacobian-type Poisson bracket) on the algebra of polynomials in evenly many variables. We show that if the base field has characteristic zero then the -ary dual coalgebra for the Jacobian -Lie algebra consists of the same linear functionals as the dual coalgebra for the commutative polynomial algebra. [ABSTRACT FROM AUTHOR]

We develop the theory of double multiplicative Poisson vertex algebras. These structures, defined at the level of associative algebras, are shown to be such that they induce a classical structure of multiplicative Poisson vertex algebra on the corresponding representation spaces. Moreover, we prove that they are in one-to-one correspondence with local lattice double Poisson algebras, a new important class among Van den Bergh's double Poisson algebras. We derive several classification results, and we exhibit their relation to non-abelian integrable differential-difference equations. A rigorous definition of double multiplicative Poisson vertex algebras in the non-local and rational cases is also provided. [ABSTRACT FROM AUTHOR]

For a standard binary branching Brownian motion on the real line, it is known that the typical value of the maximal position Mt among all particles alive at time t is mt + Θ(1) with mt = p2t - 3 2p2 log t. Further, it is proved independently in Aïdékon et al. (2013) and Arguin et al. (2013) that the branching Brownian motion shifted by mt (or Mt) converges in law to some decorated Poisson point process. The goal of this work is to study the branching Brownian motion conditioned on Mt << mt. We give a complete description of the limiting extremal process conditioned on {Mt ≤ p2αt} with α < 1, which reveals a phase transition at α = 1 - p2. We also verify the conjecture of Derrida and Shi (2017b) on the precise asymptotic behaviour of P(Mt ≤ p2αt) for α < 1. [ABSTRACT FROM AUTHOR]

Let P ( sl 2 (K)) be the Poisson enveloping algebra of the Lie algebra sl 2 (K) over an algebraically closed field K of characteristic zero. The quotient algebras P ( sl 2 (K)) / (C P − λ) , where C P is the standard Casimir element of sl 2 (K) in P ( sl 2 (K)) and 0 ≠ λ ∈ K , are proven to be simple in [U. Umirbaev and V. Zhelyabin, A Dixmier theorem for Poisson enveloping algebras, J. Algebra 568 (2021) 576–600]. Using a result by Makar–Limanov [22], we describe generators of the automorphism group of P ( sl 2 (K)) / (C P − λ) and represent this group as an amalgamated product of its subgroups. Moreover, using similar results by Dixmier [Quotients simples de l'algebre enveloppante de 2 , J. Algebra 24 (1973) 551–564] and O. Fleury [Sur les sous-groupes finis de Aut U ( 2) et Aut U () , J. Algebra 200 (1998) 404–427] for the quotient algebras U ( sl 2 (K)) / (C U − λ) , where C U is the standard Casimir element of sl 2 (K) in the universal enveloping algebra U ( sl 2 (K)) , we prove that the automorphism groups of P ( sl 2 (K)) / (C P − λ) and U ( sl 2 (K)) / (C U − λ) are isomorphic. [ABSTRACT FROM AUTHOR]

Let |$A=\boldsymbol{k}[x_1,x_2,\dots ,x_n]/I$| be a commutative algebra where |$\boldsymbol{k}$| is a field, |$\operatorname{char}(\boldsymbol{k})=0$|⁠ , and |$I\subseteq S:=\boldsymbol{k}[x_1,x_2,\dots , x_n]$| a Poisson ideal. It is well known that |$[\textrm{d} x_i,\textrm{d} x_j]:=\textrm{d}\{x_i,x_j\}$| defines a Lie bracket on the |$A$| -module |$\Omega _{A|\boldsymbol{k}}$| of Kähler differentials, making |$(A,\Omega _{A|\boldsymbol k})$| a Lie–Rinehart pair. If |$A$| is not regular, that is, |$\Omega _{A|\boldsymbol{k}}$| is not projective, the cotangent complex |$\mathbb{L}_{A|\boldsymbol{k}}$| serves as a replacement for |$\Omega _{A|\boldsymbol k}$|⁠. We prove that |$\mathbb{L}_{A|\boldsymbol{k}}$| is an |$L_\infty $| -algebroid compatible with the Lie–Rinehart pair |$(A,\Omega _{A|\boldsymbol{k}})$|⁠. The |$L_\infty $| -algebroid structure comes from a |$P_\infty $| -algebra structure on the resolvent of the morphism |$S\to A$|⁠. We identify examples when this |$L_\infty $| -algebroid simplifies to a dg Lie algebroid, concentrating on cases where |$S$| is |$\mathbb{Z}_{\ge 0}$| -graded and |$I$| and |$\{\:,\:\}$| are homogeneous. [ABSTRACT FROM AUTHOR]

Let L be a finite dimensional Lie algebra. Let U(L) and Pol(L) be the universal enveloping algebra of L and the related Poisson algebra. We find conditions for finite generation of Lie algebras [U(L),U(L)] and [Pol(L), Pol(L)]. [ABSTRACT FROM AUTHOR]

We explicitly describe the second cohomology of the Lie superalgebra 풦 ⁢ (1) of contact vector fields on the supercircle S 1 | 1 with coefficients in the spaces of weighted densities. We deduce the second cohomology of 풦 ⁢ (1) with coefficients in the Poisson algebra of pseudodifferential symbols on S 1 | 1 . We study formal deformations of the standard embedding of 풦 ⁢ (1) into the Poisson superalgebra of pseudodifferential symbols. [ABSTRACT FROM AUTHOR]

In this paper, we study conformal points among the class of E -models. The latter are σ-models formulated in terms of a current Poisson algebra, whose Lie-theoretic definition allows for a purely algebraic description of their dynamics and their 1-loop RG-flow. We use these results to formulate a simple algebraic condition on the defining data of such a model which ensures its 1-loop conformal invariance and the decoupling of its observables into two chiral Poisson algebras, describing the classical left- and right-moving fields of the theory. In the case of so-called non-degenerate E -models, these chiral sectors form two current algebras and the model takes the form of a WZW theory once realised as a σ-model. The case of degenerate E -models, in which a subalgebra of the current algebra is gauged, is more involved: the conformal condition yields a wider class of theories, which includes gauged WZW models but also other examples, seemingly different, which however sometimes turn out to be related to gauged WZW models based on other Lie algebras. For this class, we build non-local chiral fields of parafermionic-type as well as higher-spin local ones, forming classical W -algebras. In particular, we find an explicit and efficient algorithm to build these local chiral fields. These results (and their potential generalisations discussed at the end of the paper) open the way for the quantisation of a large class of conformal E -models using the standard operator formalism of two-dimensional CFT. [ABSTRACT FROM AUTHOR]