Related papers: The gl(1|1) Lie superbialgebras
We classify simple linearly compact n-Lie superalgebras with n>2 over a field F of characteristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive Z-graded Lie superalgebras…
If a Lie algebra structure g on a vector space is the sum of a family of mutually compatible Lie algebra structures g_i's, we say that g is simply assembled from the g_i's. Repeating this procedure with a number of Lie algebras, themselves…
We present a method for associating labeled directed graphs to finite-dimensional Lie algebras, thereby enabling rapid identification of key structural algebraic features. To formalize this approach, we introduce the concept of…
We investigate several matrix models based on super Lie algebras, osp(1|32,R), u(1|16,16) and gl(1|32,R). They are natural generalizations of IIB matrix model and were first proposed by Smolin. In particular, we study the supersymmetry…
This work pioneers the systematic study and classification (up to Lie algebra automorphisms) of finite-dimensional coboundary Lie bialgebras through Grassmann algebras. Several mathematical structures on Lie algebras, e.g. Killing forms or…
All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the…
The two isomorphic Borel subalgebras of gl(n), realized on upper and lower triangular matrices, allow us to consider the gl(n) \opus t_n algebra as a self-dual Drinfeld double. Compatibility conditions impose the choice of an orthonormal…
Polynomial Lie (super)algebras $g_{pd}$ are introduced via $G_{i}$-invariant polynomial Jordan maps in quantum composite models with Hamiltonians $H$ having invariance groups $G_{i}$. Algebras $g_{pd}$ have polynomial structure functions in…
We describe the Lie bialgebra structure on the Lie superalgebra sl(2,1) related to an r-matrix that cannot be obtained by a Belavin-Drinfeld type construction. This structure makes sl(2,1) into the Drinfeld double of a four-dimensional…
The Lie bialgebras of the (1+1) extended Galilei algebra are obtained and classified into four multiparametric families. Their quantum deformations are obtained, together with the corresponding deformed Casimir operators. For the coboundary…
We construct a right-invariant differential calculus on the quantum supergroup GL$_h(1| 1)$ and obtain the $h$-deformed superalgebra of GL$_h(1| 1)$.
Constructions of n-Lie algebras by strong n-Lie-Poisson algebras are given. First cohomology groups of adjoint module of Jacobian algebras are calculated. Minimal identities of 3-Jacobian algebra are found.
A new kind of graded Lie algebra (we call it $Z_{2,2}$ graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable bose subspace of the $Z_{2,2}$ graded Lie algebra and using relevant…
We propose to parametrize the configuration space of one-dimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the…
A new, so called odd Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(n|n). The related Gel'fand-Zetlin patterns are based upon the decomposition according to a particular…
We consider the variety of pre-Lie algebra structures on a given n-dimensional vector space. The group GL_n(K) acts on it, and we study the closure of the orbits with respect to the Zariski topology. This leads to the definition of pre-Lie…
The paper concerns two versions of the notion of real forms of Lie superalgebras. One is the standard approach, where a real form of a complex Lie superalgebra is a real Lie superalgebra such that its complexification is the original…
The Lie algebra gl(lambda) dependent on the complex parameter lambda is a continuous version of the Lie algebra gl(inf) of infinite matrices with only finite number of nonzero entries. The gl(lambda) was first introduced by B.L.Feigin in…
In this lecutre note, we consider infinite dimensional Lie algebras of generalized Jacobi matrices $\mathfrak{g}J(k)$ and $\mathfrak{gl}_\infty(k)$, which are important in soliton theory, and their orthogonal and symplectic subalgebras. In…
The known Holstein-Primakoff and Dyson realizations for the Lie algebras $gl(n+1),\;n=1,2,\ldots$ in terms of Bose operators are generalized to the class of the Lie superalgebras $gl(m/n+1)$ for any $n$ and $m$. Formally the expressions are…