We integrate the Lifting cocycles Ψ2n+1,Ψ2n+3,Ψ2n+5, . . . ([Sh1], [Sh2]) on the Lie algebra Difn of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle λ on an n-dimensional complex manifold M in the sense of Gelfand–Fuks cohomology [GF] (more precisely, we integrate th...

For any Lie algebra g there is a notion of Leibniz cohomology HL(g), which is defined like the classical Lie cohomology, but with the n-th tensor product g⊗n in place of the n-th exterior product Λ g. This Leibniz cohomology is defined on a larger class of algebras : the Leibniz algebras (cf. [L1], [L2]). A Leibniz algebra is a vector space equipped with a product satisfying a variation of the ...

We present a systematic construction of classical extended superconformal algebras from the hamiltonian reduction of a class of affine Lie superalgebras, which include an even subalgebra sl(2). In particular, we obtain the doubly extended N = 4 superconformal algebra Ãγ from the hamiltonian reduction of the exceptional Lie superalgebra D(2|1; γ/(1− γ)). We also find the Miura transformation for...

Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N . One calls (K,N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G := K nN which meet the annihilator k⊥ of the Lie algebra k of K do so in single K-orbits. This generalizes a result of the authors and R....

‎Let $L$ be a Lie algebra‎, ‎$mathrm{Der}(L)$ be the set of all derivations of $L$ and $mathrm{Der}_c(L)$ denote the set of all derivations $alphainmathrm{Der}(L)$ for which $alpha(x)in [x,L]:={[x,y]vert yin L}$ for all $xin L$‎. ‎We obtain an upper bound for dimension of $mathrm{Der}_c(L)$ of the finite dimensional nilpotent Lie algebra $L$ over algebraically closed fields‎. ‎Also‎, ‎we classi...

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...

In this appendix we give another algebraic proof of the Weyl character formula, using methods that have many other applications in Lie theory. We begin by setting up the machinery of Lie algebra cohomology (without assuming any previous background in homological algebra). We define the cohomology spaces for a Lie algebra representation in terms of a cochain complex and differential, with the co...

Let g be a Lie algebra of dimension n over a field K . Then g determines a multiplication table relative to each basis {e1, . . . , en}. If [ei, ej] = ∑n k=1 γ k i,jek , then (γ k i,j) ∈ K n is called a structure for g and the γ i,j the structure constants of g. The elements of Ln(K) are exactly the Lie algebra structures. They form an affine algebraic variety and the group GLn(K) acts on Ln(K)...