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...

We consider the Lie algebra of all vector fields on a contact manifold as a module over the Lie subalgebra of contact vector fields. This module is split into a direct sum of two submodules: the contact algebra itself and the space of tangent vector fields. We study the geometric nature of these two modules.

Setting up a symbolic algebraic system is the first step in mathematics mechanization of any branch of mathematics. In this paper, we establish a compact symbolic algebraic framework for local geometric computing in intrinsic differential geometry, by choosing only the Lie derivative and the covariant derivative as basic local differential operators. In this framework, not only geometric entiti...

Simple finite group schemes over an algebraically closed field of positive characteristic p 6= 2, 3 have been classified. We consider the problem of determining their infinitesimal deformations. In particular, we compute the infinitesimal deformations of the simple finite group schemes of height one corresponding to the restricted simple Lie algebras.

Let g be a complex simple Lie algebra, and h ⊂ g be a Cartan subalgebra. In the end of 1990s, B. Kostant defined two filtrations on h, one using the Clifford algebras and the odd analogue of the Harish-Chandra projection hcodd : Cl(g) → Cl(h), and the other one using the canonical isomorphism ȟ = h∗ (here ȟ is the Cartan subalgebra in the simple Lie algebra ǧ corresponding to the dual root syst...

We present the construction of an associative, commutative algebra Ĝ(X) of generalized functions on a manifold X satisfying the following optimal set of permanence properties: (i) D(X) is linearly embedded into Ĝ(X), f(p) ≡ 1 is the unity in Ĝ(X). (ii) For every smooth vector field ξ on X there exists a derivation operator L̂ξ : Ĝ(X) → Ĝ(X) which is linear and satisfies the Leibniz rule. (iii) L...

We consider the problem of decomposing a semisimple Lie algebra deened over a eld of characteristic zero as a direct sum of its simple ideals. The method is based on the decomposition of the action of a Cartan subalgebra. An implementation of the algorithm in the system ELIAS is discussed at the end of the paper.

The tipping point framework of lie detection posits that people can, and do, accurately detect deception. This framework pinpoints three circumstances that aid accuracy: (i) using methods of measurement that circumvent controlled, conscious cognition; (ii) when individual differences or situational factors portend potent risks to lie detection failure, such as in high-stakes or threatening sett...