We derive the amino acid assignment to one codon representation (typical 64-dimensional irreducible representation) of the basic classical Lie superalgebra osp(5|2) from biochemical arguments. We motivate the approach of mathematical symmetries to the classification of the building constituents of the biosphere by analogy of its success in particle physics and chemistry. The model enables to ca...

Generalized quantum statistics (GQS) associated to a Lie algebra or Lie superalgebra extends the notion of para-Bose or para-Fermi statistics. Such GQS have been classified for all classical simple Lie algebras and basic classical Lie superalgebras. In the current paper we finalize this classification for all exceptional Lie algebras and superalgebras. Since the definition of GQS is closely rel...

We continue the study of irreducible representations of the exceptional Lie superalgebra E(3, 6). This is one of the two simple infinite-dimensional Lie superalgebras of vector fields which have a Lie algebra sl(3) × sl(2) × gl(1) as the zero degree component of its consistent Z-grading. We provide the classification of the singular vectors in the degenerate Verma modules over E(3, 6), completi...

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson was established. Some non-trivial with a certain algebra (Witt algebra, $\mathcal{W}(a,-1)$, thin solvable abelian nilpotent radical) were constructed. In particular, we constructed an example the associative parts isomorphic to Laurent polynomials Witt algebra. On other side, it proven that there are no part se...

For a type I basic classical Lie superalgebra g = g0̄⊕g1̄, we establish an equivalence between typical blocks of categories of Uχ(g)-modules and Uχ(g0̄)modules. We then deduce various consequences from the equivalence.

Integrability of the one-dimensional Hubbard model and factorised scattering problem encountered on worldsheet AdS strings can be expressed in terms a peculiar quantum algebra. In this article, we derive classical limit these algebraic integrable structures based established results for exceptional simple Lie superalgebra d(2,1;epsilon) along with standard sl(2) which form supersymmetric isomet...

We show that a set of monic polynomials in the free Lie superalgebra is a Gröbner-Shirshov basis for a Lie superalgebra if and only if it is a Gröbner-Shirshov basis for its universal enveloping algebra. We investigate the structure of GröbnerShirshov bases for Kac-Moody superalgebras and give explicit constructions of Gröbner-Shirshov bases for classical Lie superalgebras. Supported in part by...

The character formula of any finite dimensional irreducible module Lλ for Lie superalgebra osp(n|2) is computed. As a by-product, the decomposition of tensor module Lλ ⊗ C n|2, where Cn|2 is the natural representation, is obtained.

We construct a new family of irreducible modules over any basic classical affine Kac-Moody Lie superalgebra which are induced from the Heisenberg subalgebra. also obtain deformations these for quantum Uqslˆ(m|n).