Related papers: Exceptional Lie Algebra $E_{7(-25)}$ (Multiplets a…
The paper presents a subclass of the class of MD5-algebras and MD5-groups, i.e. five dimensional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint representation (K-orbits) are orbits of zero or maximal…
The aim of this paper is to present remarkable classes of Lie-admissible algebras containing in particular the associative algebras, the Vinberg algebras and pre-Lie algebras. We determine the associated quadratic operads and their dual…
We propose the method for obtaining invariants of arbitrary representations of Lie groups that reduces this problem to known problems of linear algebra. The basis of this method is the idea of a special extension of the representation…
We study the phenomena that arise when we combine the standard pseudodifferential operators with those operators that appear in the study of some sub-elliptic estimates, and on strongly pseudoconvex domains. The algebra of operators we…
We study realizations of polynomial deformations of the sl(2,R)- Lie algebra in terms of differential operators strongly related to bosonic operators. We also distinguish their finite- and infinite-dimensional representations. The linear,…
For the exceptional finite-dimensional modular Lie superalgebras $\mathfrak{g}(A)$ with indecomposable Cartan matrix $A$, and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd…
The spectral eta-invariant of a self-adjoint elliptic differential operator on a closed manifold is rigid, provided that the parity of the order is opposite to the parity of dimension of the manifold. The paper deals with the calculation of…
In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of $E_7$ into a sum of irreducible submodules. Moreover, we obtain a combinatorial identity, saying…
To a finite dimensional representation of a complex Lie group $G$, an associative algebra of adjoint covariant polynomial maps from the direct sum of $m$ copies of the Lie algebra $\mathfrak{g}$ of $G$ into an algebra of complex matrices is…
In recent years, averaging operators on Lie algebras (also called embedding tensors in the physics literature) and associated tensor hierarchies form an efficient tool for constructing supergravity and higher gauge theories. A Lie algebra…
Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The corresponding equivariant map algebra is the Lie algebra M of equivariant regular maps from X to g. We classify the irreducible finite-dimensional…
We fill in the "hole" in the exceptional series of Lie algebras that was observed by Cvitanovic, Deligne, Cohen and deMan. More precisely, we show that the intermediate Lie algebra between $E_7$ and $E_8$ satisfies some of the decomposition…
We compute all complex structures on indecomposable 6-dimensional real Lie algebras and their equivalence classes. We also give for each of them a global holomorphic chart on the connected simply connected Lie group associated to the real…
Certain infinite families of operator identities related to powers of positive root generators of (super) Lie algebras of first-order differential operators and $q$-deformed algebras of first-order finite-difference operators are presented.
Logarithmic representations of the conformal Galilean algebra (CGA) and the Exotic Conformal Galilean algebra ({\sc ecga}) are constructed. This can be achieved by non-decomposable representations of the scaling dimensions or the rapidity…
We study families of self-adjoint operators with given spectra whose sum is a scalar operator. Such families are $*$-representations of certain algebras which can be described in terms of graphs and positive functions on them. The main…
First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to…
For any skew symmetric matrix over complex numbers, we introduce an EALA and it is called Skew Symmetric Extended Affine Lie Algebra (SSEALA). This way we get a large class of EALAs and most often they are non-isomorphic. In this paper we…
The indecomposable solvable Lie algebras with graded nilradical of maximal nilindex and a Heisenberg subalgebra of codimension one are analyzed, and their generalized Casimir invariants calculated. It is shown that rank one solvable…
A non completely reducible symplectic Lie algebra is a symplectic Lie algebra which cannot be symplectically reduced to the trivial symplectic Lie algebra. Our aim is to provide a complete classification, up to symplectomorphism of non…