Twisted Rota–Baxter operators and Reynolds operators on Lie algebras and NS-Lie algebras

Family of operators and their Lie algebras

Lie Algebras of Differential Operators and Lie-Algebraic Potentials

An explicit characterisation of all second order differential operators on the line which can be written as bilinear combinations of the generators of a linitedimensional Lie algebra of first order differential operators is found, solving a problem arising in the Lie-algebraic approach to scattering theory and molecular dynamics. One-dimensional potentials corresponding to these Lie algebras ar...

Lie $^*$-double derivations on Lie $C^*$-algebras

A unital $C^*$ -- algebra $mathcal A,$ endowed withthe Lie product $[x,y]=xy- yx$ on $mathcal A,$ is called a Lie$C^*$ -- algebra. Let $mathcal A$ be a Lie $C^*$ -- algebra and$g,h:mathcal A to mathcal A$ be $Bbb C$ -- linear mappings. A$Bbb C$ -- linear mapping $f:mathcal A to mathcal A$ is calleda Lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all ...

Twisted Toroidal Lie Algebras

Using n finite order automorphisms on a simple complex Lie algebra we construct twisted n-toroidal Lie algebras. Thus obtaining Lie algebras wich have a rootspace decomposition. For the case n = 2 we list certain simple Lie algebras and their automorphisms, which produce twisted 2-toroidal algebras. In this way we obtain Lie algebras that are related to all Extended Affine Root Systems of K. Sa...

Representations of Lie Algebras by Normal Operators

Recently I. E. Segal [2] has proposed a study of the unitary representations of a complex semisimple Lie group G by studying "analytic" holomorphic representations of G by normal operators. To this end he proved that every unitary representation U of G may be written U(g)=R(g)R(g~1)* (gEG) where R is an analytic holomorphic representation of G by normal operators such that if R(gi) and R(g2) ar...