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Classical limits of quantum groups give rise to multiplicative Poisson structures such as Poisson-Lie and quasi-Poisson structures. We relate them to the notion of a shifted Poisson structure which gives a conceptual framework for…
Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the $\nabla u=f(u)$ Poisson's equation, which has a subalgebra isomorphic to the $3-$dimensional special Euclidean group ${\rm SE}(3)$ or group of…
In this paper, we study invariant Poisson structures on homogeneous manifolds, which serve as a natural generalization of homogeneous symplectic manifolds previously explored in the literature. Our work begins by providing an algebraic…
In this paper, we study the Lie point symmetry group of a system describing an ideal plastic plane flow in two dimensions in order to find analytical solutions of the system. The infinitesimal generators that span the Lie algebra for this…
We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation…
Arnold pointed out that the Euler equation of incompressible ideal hydrodynamics describes geodesics on the group of volume-preserving diffeomorphisms. A simple analogue is the Euler equation for a rigid body, which is the geodesic equation…
Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures…
We derive a formula for the the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular…
In this note the notion of Poisson brackets in Kontsevich's "Lie World" is developed. These brackets can be thought of as "universally" defined classical Poisson structures, namely formal expressions only involving the structure maps of a…
Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a…
Classical algebraic structures require exact satisfaction of their defining axioms. We propose similarity algebra, a framework extending algebraic and Lie structures to settings where operations satisfy quantitative bounds up to a tolerance…
The Lie-point symmetry method is used to find some closed-form solutions for a constitutive equation modeling stress in elastic materials. The partial differential equation (PDE), which involves a power law with arbitrary exponent n, was…
The theory of multidimensional Poisson vertex algebras (mPVAs) provides a completely algebraic formalism to study the Hamiltonian structure of PDEs, for any number of dependent and independent variables. In this paper, we compute the…
We examine the Wess-Zumino-Novikov-Witten (WZNW) model on a circle and compute the Poisson bracket algebra for left and right moving chiral group elements. Our computations apply for arbitrary groups and boundary conditions, the latter…
The Poisson gauge theory is a semi-classical limit of full non-commutative gauge theory. In this work we construct an L$_\infty^{full}$ algebra which governs both the action of gauge symmetries and the dynamics of the Poisson gauge theory.…
We study a Lie algebra $\mathcal A_{a_1,\ldots,a_{n-1}}$ of deformed skew-symmetric $n \times n$ matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations are parametrized by a sequence of real…
The use of discrete material representation in numerical models is advantageous due to the straightforward way it takes into account material heterogeneity and randomness, and the discrete and orientated nature of cracks. Unfortunately, it…
In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very "ascetic" in that only the structure of a locally compact topological group was used.…
A Riemann-Lie algebra is a Lie algebra $\cal G$ such that its dual ${\cal G}^*$ carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, S\'erie I, (2001) 763-768) with the canonical linear…
The "coquecigrue" problem for Leibniz algebras is that of finding an appropriate generalization of Lie's third theorem, that is, of finding a generalization of the notion of group such that Leibniz algebras are the corresponding tangent…