In this work, the Z3-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algebra and the dual algebra is given. E-mail: [email protected]

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G|+ 1, where |G| is the order of the commutator subgroup. The authors have previously determined the groups G for which this index is maximal and here they determine the G for which it is ‘almost maximal’, that ...

The construction of a class of associative composition algebras qn on R 4 generalizing the wellknown quaternions Q provides an explicit representation of the universal enveloping algebra of the real three-dimensional Lie algebras having tracefree adjoint representations (class A Bianchi type Lie algebras). The identity components of the four-dimensional Lie groups GL(qn,l) Cqn (general linear g...

The main goal of this paper is to give some construction results BiHom-post-Lie algebras which are a generalization both post-Lie-algebras and Hom-post-Lie algebras. They the algebraic structures behind weighted $${\mathcal {O}}$$ -operator BiHom-Lie can be also regarded as splitting into three parts structure BiHom-Lie-algebra. Moreover we develop representation theory on vector space V. We sh...

is called a truncated current Lie algebra, or sometimes a generalised Takiff algebra. We shall describe a highest-weight theory for ĝ, and the reducibility criterion for the universal objects of this theory, the Verma modules. The principal motivation, beside the aesthetic, is that certain representations of affine Lie algebras are essentially representations of a truncated current Lie algebra....

The discrete symmetries P, C and T are discussed in terms of Lie algebra extensions of the Poincare Lie algebra. This formulation leads to certain problems of Lie algebra theory which will be presented in this and succeeding papers. (To be submitted to Communications in Mathematical Physics) * Work supported by the U. S. Atomic Energy Commission.

Abstract. In this paper we investigate Lie bialgebra structures on a twisted Schrödinger-Virasoro type algebra L. All Lie bialgebra structures on L are triangular coboundary, which is different from the relative result on the original Schrödinger-Virasoro type Lie algebra. In particular, we find for this Lie algebra that there are more hidden inner derivations from itself to L⊗L and we develop ...

<p style='text-indent:20px;'>In this paper, we characterize the graded post-Lie algebra structures on Schrödinger-Virasoro Lie algebra. Furthermore, as an application, obtain all homogeneous Rota-Baxter operator of weight <inline-formula><tex-math id="M1">$ 1 $</tex-math></inline-formula> algebra.</p>

We consider the universal central extension of the Lie algebra Vect(S 1)⋉C ∞ (S 1). The coadjoint representation of this Lie algebra has a natural geometric interpretation by matrix analogues of the Sturm-Liouville operators. This approach leads to new Lie superalgebras generalizing the well-known Neveu-Schwartz algebra.

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group G, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of G. MSC 2000 : 20C15,17B99.