Related papers: $\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-gra…
We construct Lie algebras arising from cubic norm pairs over arbitrary commutative base rings. Such Lie algebras admit a grading by a root system of type $G_2$, and when the cubic norm pair is a cubic Jordan matrix algebra, the…
Let $\mathbb{K}$ be an algebraically closed field of characteristic 0. A finite dimensional Lie algebra $\mathfrak{g}$ over $\mathbb{K}$ is said to be stable if there exists a linear form $g\in\mathfrak{g}^{*}$ and a Zariski open subset in…
To provide tools, especially L-operators, for use in studies of rational Yang-Baxter algebras and quantum integrable models when the Lie algebras so(N) (b_n, d_n) or sp(2n) (c_n) are the invariance algebras of their R matrices, this paper…
We give an algebraic construction of the topological graph-tree configuration pairing of Sinha and Walter beginning with the classical presentation of Lie coalgebras via coefficients of words in the associative Lie polynomial. Our work…
In the present paper we continue the project of systematic classification and construction of invariant differential operators for non-compact semisimple Lie groups. This time we make the stress on one of the main building blocks, namely…
The generators and commutation relations are calculated explicitly for higher symmetry algebras of a class of hyperbolic Euler-Lagrange systems of Liouville type (in particular, for 2D Toda chains associated with semi-simple complex Lie…
In [7, Papadima and Suciu, When does the associated graded Lie algebra of an arrangement group decompose? Comment. Math. Helv. {\bf 81:4} (2006), 859--875] it is proved that the holonomy Lie algebra of an arrangement of hyperplanes through…
We construct a new family of infinite-dimensional quasi-graded Lie algebras on hyperelliptic curves. We show that constructed algebras possess infinite number of invariant functions and admit a decomposition into the direct sum of two…
This paper is devoted to investigating the structure theory of a class of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras. In particular, we completely determine the derivation algebras, the automorphism…
We present a special and attractive basis for the exceptional Lie algebra $G_2$, which turns $G_2$ into a $\mathbb{Z}_2^3$-graded Lie algebra. There are two basis elements for each degree of $\mathbb{Z}_2^3\setminus\{(0,0,0)\}$, thus…
We present a framework for enlarging the construction of $\mathbb{Z}_2^2$-graded classical Toda theory from the class of $\mathbb{Z}_2^2$-graded Lie algebras to the class of $\mathbb{Z}_2^2$-graded Lie superalgebras. This scheme is applied…
In one of his last papers, Boris Weisfeiler proved that if modular semisimple Lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration, then associated graded algebra is isomorphic to one constructed from…
We review various generalizations of supersymmetry and discuss their relationship. In particular, we show how supersymmetry, parasupersymmetry, fractional supersymmetry, orthosupersymmetry, and the Z_n-graded topological symmetries are…
Definitions of the parastatistics algebras and known results on their Lie (super)algebraic structure are reviewed. The notion of super-Hopf algebra is discussed. The bosonisation technique for switching a Hopf algebra in a braided category…
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…
In this paper we reexamine the definition of parafermions and parabosons by means of Green's triple relations, and extend these relations by including a parity operator $P$ which is also determined by means of triple relations. As a…
Gabriel's Theorem, and the work of Bernstein, Gelfand and Ponomarev established a connection between the theory of quiver representations and the theory of simple Lie algebras. Lie superalgebras have been studied from many perspectives, and…
We show that our construction of realizations for Lie algebras and quantum algebras can be generalized to quantum superalgebras, too. We study an example of quantum superalgebra $U_q(gl(2/1))$ and give the boson-fermion realization with…
We introduce the notion of G-algebroid, generalising both Lie and Courant algebroids, as well as the algebroids used in $E_{n(n)}\times\mathbb{R}^+$ exceptional generalised geometry for $n\in\{3,\dots,6\}$. Focusing on the exceptional case,…
It is known that the defining triple relations of m pairs of parafermion operators and n pairs of paraboson operators with relative parafermion relations can be considered as defining relations for the Lie superalgebra osp(2m+1|2n) in terms…