is surjective. If W is a Q-vector space, Q-algebra or Q-Lie algebra then we write WC for the correspondingC-vector space (respectively, C-algebra orC-Lie algebra) W ⊗Q C. Let f(x) ∈ C[x] be a polynomial of degree n ≥ 2 without multiple roots. Suppose that p is a prime that does not divide n and a positive integer q = p is a power of p. As usual, φ(q) = (p − 1)p denotes the Euler function. Let u...

Let Sn = K[[x1, . . . , xn]] be the algebra of power series over a field K of characteristic zero, S c n be the group of continuous automorphisms of Sn with constant Jacobian, and Div c n be the Lie algebra of derivations of Sn with constant divergence. We prove that AutLie(Div c n) = AutLie,c(Div c n) ≃ S c n.

Let G be a connected linear semisimple Lie group with Lie algebra g, and let K C → Aut(p C ) be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in g and O is the nilpotent K C -orbit in p C associated to Ω by the Kostant-Sekiguchi correspondence. We show that the complexity of O as a K C variety measures ...

A current Lie algebra is contructed from a tensor product of a Lie algebra and a commutative associative algebra of dimension greater than 2. In this work we are interested in deformations of such algebras and in the problem of rigidity. In particular we prove that a current Lie algebra is rigid if it is isomorphic to a direct product g× g × ...× g where g is a rigid Lie algebra. 1 Current Lie ...

Let G be a simply connected connected real nilpotent Lie group with Lie algebra g, H a connected closed subgroup of G with Lie algebra h and β ∈ h∗ satisfying β([h, h]) = {0}. Let χβ be the unitary character of H with differential 2 √ −1πβ at the origin. Let τ ≡ IndHχβ be the unitary representation of G induced from the character χβ of H. We consider the algebra D(G,H, β) of differential operat...

In his article [3] in the Mathematical Intelligencer (vol. 11, no. 3) A. John Coleman gives a colorful biography of the mathematician Wilhelm Killing and offers an admiring appraisal of his paper [8]. The subject of this paper, the classification of the simple Lie algebras over C, has indeed turned out to be a milestone in the history of mathematics. At the conclusion of his article Coleman lis...

For each simple Lie algebra g consider the corresponding affine vertex algebra Vcrit(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A fo...

In this work we consider 2-step nilradicals of parabolic subalgebras the simple Lie algebra An and describe a new family faithful nil-representations na,c, a,c ? N. We obtain sharp upper bound for minimal dimension ?(na,c) several pairs (a,c) ?(na,c).

1. Lecture 1: Affine Lie algebras and the Fock representation of ĝln. 1.1. The loop algebra construction. Let g be a complex reductive Lie algebra and let L denote the algebra of Laurent polynomials in one variable L = C[t, t−1]. The loop algebra over g is L(g) = L ⊗ g, which is a Lie algebra with the bracket [t ⊗ x, t ⊗ y]0 = t[x, y]. (1.1) The elements of the loop algebra may be regarded as r...