In this paper, first we introduce the notion of a nonabelian embedding tensor, which is generalization an tensor. Then Leibniz-Lie algebra, underlying algebraic structure and can also be viewed as Leibniz algebra. Next using derived bracket, construct differential graded Lie whose Maurer-Cartan elements are exactly tensors. Consequently, obtain algebra that governs deformations Finally, define ...

Ghost neutrino solution in radiative Kerr spacetime endowed with totally skew-symmetric Cartan contortion is presented. The computations are made by using the Newman-Penrose (NP) calculus. The model discussed here maybe useful in several astrophysical applications specially in black hole astrophysics. PACS number(s): 0420, 0450 Departamento de F́ısica Teorica-UERJ. Rua São Fco. Xavier 524, Rio d...

We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A ⊂ M , the Betti numbers of the standard equivalence relation associated with A ⊂ M ([G2]), are in fact isomorphism invariants for the factorsM , β HT...

We prove that the normalizer of any diffuse amenable subalgebra of a free group factor L(Fr) generates an amenable von Neumann subalgebra. Moreover, any II1 factor of the form Q ⊗̄L(Fr), with Q an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group Fr, 2 ≤ r ≤ ∞, on a probability ...

The classical Cartan’s structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent bundle with a symmetric bilinear form which is allowed to become degenerate. B...

We investigate the Cartan formalism in F(R) gravity. gravity has been introduced as a theory to explain cosmologically accelerated expansions by replacing Ricci scalar R Einstein–Hilbert action with function of R. As is well-known, rewritten scalar–tensor using conformal transformation. described based on Riemann–Cartan geometry formulated vierbein-associated local Lorenz symmetry. In formalism...

It is shown that for each 0 < lambda < 1, the free Araki-Woods factor of type III(lambda) cannot be written as a tensor product of two diffuse von Neumann algebras (i.e., is prime) and does not contain a Cartan subalgebra.

• Euclid synthetic geometry 300 BC • Descartes analytic geometry 1637 • Gauss – complex algebra 1798 • Hamilton – quaternions 1843 • Grassmann – Grasmann Algebra 1844 • Cayley – Matrix Algebra 1854 • Clifford – Clifford algebra 1878 • Gibbs – vector calculus 1881 – used today • Sylvester – determinants 1878 • Ricci – tensor calculus 1890 • Cartan – differential forms 1908 • Dirac, Pauli – spin ...

First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section 6 we generalize the Maurer-Cartan equation to strongly homotopy Li...