Uncertainty principles generated by Lie-groups Abstract: For unbounded operators A, B and C in general, the commutation relation [A, B] = C does not lead to the uncertainty relation ‖Au‖‖Bu‖ ≥ 1 2 |〈Cu, u〉|. If A, B and C are part of the generators of a unitary representation of a Lie group then the uncertainty principle above holds.

The multiplication in a quasiassociative algebra R satisfies the property a ∗ (b ∗ c)− (a ∗ b) ∗ c = b ∗ (a ∗ c)− (b ∗ a) ∗ c, a, b, c ∈ R. (∗∗) This property is necessary and sufficient for the Lie algebra Lie(R) to have a phase space. The above formulae are put into a cohomological framework, with the relevant complex being different from the Hochschild one even when the relevant quasiassocia...

Assume that $(N,L)$, is a pair of finite dimensional nilpotent Lie algebras, in which $L$ is non-abelian and $N$ is an ideal in $L$ and also $mathcal{M}(N,L)$ is the Schur multiplier of the pair $(N,L)$. Motivated by characterization of the pairs $(N,L)$ of finite dimensional nilpotent Lie algebras by their Schur multipliers (Arabyani, et al. 2014) we prove some properties of a pair of nilpoten...

We associate to any germ of an analytic variety a Lie algebra of tangent vector fields, the tangent algebra. Conversely, to any Lie algebra of vector fields an analytic germ can be associated, the integral variety. The paper investigates properties of this correspondence: The set of all tangent algebras is characterized in purely Lie algebra theoretic terms. And it is shown that the tangent alg...

Let hom-Ln(C) be the algebraic set of complex n-dimensional hom-Lie algebras. The group GL(n,C) acts on it via change basis. We classify structures with nilpotent twisting map 3-dimensional Lie algebras, up to isomorphism, and we give classification orbit closures in such family. For this purpose, introduce some invariants algebras study both problems. ideas techniques presented here can easily...

‎in this paper‎, ‎we introduce the structure of a groupoid associated to a vector field‎ ‎on a smooth manifold‎. ‎we show that in the case of the $1$-dimensional manifolds‎, ‎our‎ ‎groupoid has a‎ ‎smooth structure such that makes it into a lie groupoid‎. ‎using this approach‎, ‎we associated to‎ ‎every vector field an equivalence‎ ‎relation on the lie algebra of all vector fields on the smooth...

A nonzero 2-cocycle Γ ∈ Z(g,R) on the Lie algebra g of a compact Lie group G defines a twisted version of the Lie-Poisson structure on the dual Lie algebra g∗, leading to a Poisson algebra C∞(g∗(Γ)). Similarly, a multiplier c ∈ Z (G, U(1)) on G which is smooth near the identity defines a twist in the convolution product on G, encoded by the twisted group C∗algebra C∗(G, c). Further to some supe...

We introduce the notion of a Poisson symmetric space and the associated infinitesimal object, a symmetric Lie bialgebra. They generalize corresponding notions for Lie groups due to V. G. Drinfel’d. We use them to give some geometric insight to certain Poisson brackets that have appeared before in the literature. 1 Motivation Let us recall briefly the best-known examples of Poisson manifolds. Th...