In the two papers of this series, we initiate development a new approach to implementing concept symmetry in classical field theory, based on replacing Lie groups/algebras by groupoids/algebroids, which are appropriate mathematical tools describe local symmetries when gauge transformations combined with space-time transformations. second part, shall adapt formalism developed first paper context...

B-series originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge–Kutta methods. Connections to renormalization have been established in recent years. The algebraic structure of classical Runge–Kutta methods is described by the Connes–Kreimer Hopf algebra. Lie–Butcher theory is a generalization of B-series ai...

A Lie algebraic approach to the unitary transformations in Weyl quantization is discussed. This approach, being formally equivalent to the ⋆-quantization is an extension of the classical Poisson-Lie formalism which can be used as an efficient tool in the quantum phase space transformation theory. The purpose of this paper is to show that the Weyl correspondence in the quantum phase space can be...

This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it ...

n root of a Lie algebra and its dual (that is fractional supergroup ) based on the permutation group Sn invariant forms is formulated in the Hopf algebra formalism. Detailed discussion of S3-graded sl(2) algebras is done.

Wigner distributions for quantum mechanical systems whose configuration space is a finite group of odd order are defined so that they correctly reproduce the marginals and have desirable transformation properties under left and right translations. While for the Abelian case we recover known results, though from a different perspective, for the non Abelian case, our results appear to be new. ema...

1 Classical Lie algebras A Lie algebra is a vector space g with a bilinear map [, ] : g× g → g such that (a) [x, y] = −[y, x], for x, y ∈ g, and (b) (Jacobi identity) [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0, for all x, y, z ∈ g. A bilinear form 〈, 〉 : g× g → C is ad-invariant if, for all x, y, z ∈ g, 〈adx(y), z〉 = −〈y, adx(z)〉, where adx(y) = [x, y], (1.1) for x, y,∈ g. The Killing form is ...

We have previously described an embedding of the Poincaré Lie algebra into an extension of the Lie field of the group SO0(1, 4), and we used this embedding to construct irreducible representations of the Poincaré group out of representations of SO0(1, 4). Some q generalizations of these results have been obtained by us i.e. we embed classical structures into quantum structures. Here we report o...

We study the double cosets of a Lie group by a compact Lie subgroup. We show that a Weil formula holds for double coset Lie hypergroups and show that certain representations of the Lie group lift to representations of the double coset Lie hypergroup. We characterize smooth (analytic) vectors of these lifted representations.