Related papers: Group Contractions via Infinite-Dimensional Lie Th…
After classifying indecomposable quasi-classical Lie algebras in low dimension, and showing the existence of non-reductive stable quasi-classical Lie algebras, we focus on the problem of obtaining sufficient conditions for a quasi-classical…
In this series of two papers we will generalise the concept of extending a Lie algebroid by a Lie algebra bundle, leading to a notion of extending a Lie algebroid by another Lie algebroid whose orbits lie in the orbits of the former…
Nongraded infinite-dimensional Lie algebras appeared naturally in the theory of Hamiltonian operators, the theory of vertex algebras and their multi-variable analogues. They play important roles in mathematical physics. This survey article…
This is a short presentation of some classical results on finite dimensional complex Lie algebras (classification of nilpotent Lie algebras, deformations and perturbations, contractions and rigidity). We present some applications to…
We propose a general framework to contract unitary dual of Lie groups via holomorphic quantization of their co-adjoint orbits. The sufficient condition for the contractability of a representation is expressed via cocycles on coadjoint…
Formal actions of Lie algebras over vector spaces are introduced in a purely algebraic way, as a mimic of infinitesimal operations of Banach Lie algebras over Banach analytic manifolds. In analogy with the case of abstract groups, complete…
We provide a framework for extensions of Lie algebroids, including non-abelian extensions and Lie algebroids over different bases. Our approach involves Ehresmann connections, which allows straight generalizations of classical…
We consider the group algebra over the field of complex numbers of the Weyl group of type B (the hyperoctahedral group, or the group of signed permutations) and of the Weyl group of type D (the demihyperoctahedral group, or the group of…
We call a unital locally convex algebra $A$ a continuous inverse algebra if its unit group $A^\times$ is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group $G$ on a…
Starting from gravity as a Chern-Simons action for the AdS algebra in five dimensions, it is possible to deform the theory through an expansion of the Lie algebra that leads to a system consisting of the Einstein-Hilbert action plus…
Generalizations in many directions of the contraction procedure for Lie algebras introduced by E.J.Saletan are proposed. Products of arbitrary nature, not necessarily Lie brackets, are considered on sections of finite-dimensional vector…
Every Lie algebra over a field $E$ gives rise to new Lie algebras over any subfield $F \subseteq E$ by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of…
To a Lie groupoid over a compact base, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present…
This paper is about the relation of the geometry of Lie groupoids over a fixed compact manifold and the geometry of their (infinite-dimensional) bisection Lie groups. In the first part of the paper we investigate the relation of the…
We introduce braided Lie bialgebras as the infinitesimal version of braided groups. They are Lie algebras and Lie coalgebras with the coboundary of the Lie cobracket an infinitesimal braiding. We provide theorems of transmutation, Lie…
We prove that there exists just one pair of complex four-dimensional Lie algebras such that a well-defined contraction among them is not equivalent to a generalized IW-contraction (or to a one-parametric subgroup degeneration in…
An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists! \land p = q$. For a logic $L$ algebraized by a quasivariety $\mathcal{Q}$ we show that the AE-subclasses of…
We study tensors on Lie groupoids suitably compatible with the groupoid structure, called {\em multiplicative}. Our main result gives a complete description of these objects only in terms of infinitesimal data. Special cases include the…
We develop Lie theory in the category $\text{Ver}_4^+$ over a field of characteristic 2, the simplest tensor category which is not Frobenius exact, as a continuation of arXiv:2406.10201. We provide a conceptual proof that an operadic Lie…
Quantum theory can be formulated as a theory of operations, more specific, of complex represented operations from real Lie groups. Hilbert space eigenvectors of acting Lie operations are used as states or particles. The simplest simple Lie…