Let $L$ be a finite-dimensional Lie algebra. We say a subalgebra $H$ of $L$ is permutably complemented in $L$ if there is a subalgebra $K$ of $L$ such that $L=H+K$ and $Hcap K=0$. Also, if every subalgebra of $L$ is permutably complemented in $L$, then $L$ is called completely factorisable. In this article, we consider the influence of these concepts on the structure of a Lie algebra, in partic...

Abstract We realize the simple Lie superalgebra G ( 3 ) as supersymmetry of various geometric structures, most importantly super-versions Hilbert–Cartan equation (SHC) and Cartan's involutive PDE system that exhibit 2 symmetry. provide symmetries explicitly compute, via first Spencer cohomology groups, Tanaka–Weisfeiler prolongation negatively graded superalgebras associated with two particular...

We study the eigenspace decomposition of a basic classical Lie superalgebra under adjoint action toral subalgebra, thus extending results Kostant. In recognition Kostant's contribution we refer to eigenfunctions appearing in as Kostant roots. then prove that root systems inherit main properties systems. Our approach is combinatorial nature and utilizes certain graphs naturally associated with p...

An explicit construction of all finite-dimensional irreducible representations of the Lie superalgebra gl(1|n) in a Gel’fand-Zetlin basis is given. The notion of Wigner Quantum Oscillators (WQOs) is recalled. The star type I representations of gl(1|n) are physical state spaces of the WQO. These solutions have remarkable properties following from the spectrum of the Hamiltonian and of the positi...

We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of prime characteristic and announce that the classification of all finite dimensional simple Lie algebras over an algebraically closed field of characteristic p > 3 is now complete. Any such Lie algebra is up to isomorphism either classical or a filtered Lie algebra of Cartan type...

Our main purpose is to develop the theory of existence of superantiautomorphisms of order two of the second kind (which are caled superinvolutions of the second kind) on finite dimensional central simple superalgebras A = Mn(D) , where D is a finite dimensional division superalgebra with nontrivial grading over K , where K is a field of any characteristic. We determine which finite dimensional ...

In our recent paper [DSK11] we computed the dimension of the variational Poisson cohomology H•K(V) for any quasiconstant coefficient l × l matrix differential operator K of order N with invertible leading coefficient, provided that V is a normal algebra of differential functions over a linearly closed differential field. In the present paper we show that, for K skewadjoint, the Z-graded Lie sup...

In our recent paper [DSK11] we computed the dimension of the variational Poisson cohomology H•K(V) for any quasiconstant coefficient l × l matrix differential operator K of order N with invertible leading coefficient, provided that V is a normal algebra of differential functions over a linearly closed differential field. In the present paper we show that, for K skewadjoint, the Z-graded Lie sup...

We study infinite-dimensional analogs of classical Lie superalgebras over an algebraically closed field F zero characteristic. Let I be infinite set. For algebra M_∞ (I) × matrices a ground having finitely many nonzero entries, we consider the related superalgebra gl_∞ (I1, I2) and its commutator sl_∞ for disjoint union nonempty subsets I1 I2 set I; describe derivations I2).