6 Lie Derivatives and the Commutator Revisited 21 6.1 Integral Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.2 Congruence of Curves . . . . . . . . . . . . . . . . . . . . . . . 21 6.3 The Commutator Revisited: A Geometric Interpretation . . . 22 6.4 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.5 Lie Derivatives of a Function . . . . . . . . . . . ....

 Motivated by the intensive and powerful works concerning additive‎ ‎mappings of operator algebras‎, ‎we mainly study Lie-type higher‎ ‎derivations on operator algebras in the current work‎. ‎It is shown‎ ‎that every Lie (triple-)higher derivation on some classical operator‎ ‎algebras is of standard form‎. ‎The definition of Lie $n$-higher‎ ‎derivations on operator algebras and related pot...

It is shown that the problem of reduction can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of these algebras, beyond the context of integrable systems. Moreover, it is proven that sl2(C)–Automorphic Lie Algebras associated to the icosahedral group I, the octahedral group O, the tetrahedral grou...

We discovered an over-density of H-alpha-emitting galaxies associated with a Planck compact source in the COSMOS field (PHzG237.0+42.5) through narrow-band imaging observations Subaru/MOIRCS. This Planck-selected dusty proto-cluster at z=2.16 has 38 H-alpha emitters including six spectroscopically confirmed observed MOIRCS 4'x7' (corresponding to ~2.0x3.5~Mpc^2 physical scale). find that massiv...

We give a complete classification of infinite dimensional indecomposable weight modules over the Lie superalgebra sl(2/1). §1. Introduction Among the basic-classical Lie superalgebras classified by Kac [3], the lowest dimensional of these is the Lie superalgebra B(0, 1) or osp(1, 2), while the lowest dimensional of these which has an isotropic odd simple root is the Lie superalgebra A(1, 0) or ...

We consider the universal central extension of the Lie algebra Vect(S 1)⋉C ∞ (S 1). The coadjoint representation of this Lie algebra has a natural geometric interpretation by matrix analogues of the Sturm-Liouville operators. This approach leads to new Lie superalgebras generalizing the well-known Neveu-Schwartz algebra.

We prove a Poincaré lemma for a set of r smooth functions on a 2n-dimensional smooth manifold satisfying a commutation relation determined by r singular vector fields associated to a Cartan subalgebra of sp(2r, R). This result has a natural interpretation in terms of the cohomology associated to the infinitesimal deformation of a completely integrable system.

In this note we compute Leibniz algebra deformations of the 3-dimensional nilpotent Lie algebra n3 and compare it with its Lie deformations. It turns out that there are 3 extra Leibniz deformations. We also describe the versal Leibniz deformation of n3 with the versal base.

In "Functional MRI Lie Detection: Too Good to be True?" in this issue of The Journal, Joseph Simpson reviews the merits and the limitations of using fMRI to detect deception. After presenting the gaps in experimental data that stand in the way of translating the laboratory proof of concept to a field application, Simpson surveys the legal, regulatory and ethics concerns facing fMRI, should it e...