Domestic canonical algebras and simple Lie algebras

For each simply-laced Dynkin graph Δ we realize the simple complex Lie algebra of type Δ as a quotient algebra of the complex degenerate composition Lie algebra of a domestic canonical algebra A of type Δ by some ideal I of that is defined via the Hall algebra of A, and give an explicit form of I. Moreover, we show that each root space of has a basis given by the coset of an indecomposable A-module M with root easily computed by the dimension vector of M. Dedicated to Professor Claus Michael Ringel on the occasion of his 60th birthday.     

Complex finitary simple Lie algebras

An algebra is called finitary if it consists of finite-rank transformations of a vector space. We classify finitary simple Lie algebras over an algebraically closed field of zero characteristic. It is shown that any such algebra is isomorphic to one of the following¶ (1) a special transvection algebra \frak t(V,P)\frak t(V,\mit\Pi );¶ (2) a finitary orthogonal algebra \frak fso (V,q)\frak {fso} (V,q); ¶ (3) a finitary symplectic algebra \frak fsp (V,s)\frak {fsp} (V,s).¶Here V is an infinite dimensional K-space; q (respectively, s) is a symmetric (respectively, skew-symmetric) nondegenerate bilinear form on V; and P\Pi is a subspace of the dual V^* whose annihilator in V is trivial: 0={v ? V | Pv=0}0=\{{v}\in V\mid \Pi {v}=0\}.     

On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

We prove a part of the Cachazo-Douglas-Seiberg-Witten conjecture uniformly for any simple Lie algebra . The main ingredients in the proof are: Garland's result on the Lie algebra cohomology of ; Kostant's result on the `diagonal' cohomolgy of and its connection with abelian ideals in a Borel subalgebra of ; and a certain deformation of the singular cohomology of the infinite Grassmannian introduced by Belkale-Kumar.

     

Triple automorphisms of simple Lie algebras

An invertible linear map φ on a Lie algebra L is called a triple automorphism of it if φ([x, [y, z]]) = [φ(x), [φ(y), φ(z)]] for ∀ x, y, z ∈ L. Let g be a finite-dimensional simple Lie algebra of rank l defined over an algebraically closed field F of characteristic zero, p an arbitrary parabolic subalgebra of g. It is shown in this paper that an invertible linear map φ on p is a triple automorphism if and only if either φ itself is an automorphism of p or it is the composition of an automorphism of p and an extremal map of order 2.     

Finitely presented infinite-dimensional simple Lie algebras

Archiv der Mathematik -     

Construction and classification of complex simple Lie algebras via projective geometry

We construct the complex simple Lie algebras using elementary algebraic geometry. We use our construction to obtain a new proof of the classification of complex simple Lie algebras that does not appeal to the classification of root systems.     

Classification of simple cuspidal modules for solenoidal Lie algebras

We give a new conceptual proof of the classification of cuspidal modules for the solenoidal Lie algebra. This classification was originally published by Y. Su in 2001. Our proof is based on the theory of modules for the solenoidal Lie algebras that admit a compatible action of the commutative algebra of functions on a torus.     

A universal dimension formula for complex simple Lie algebras

We present a universal formula for the dimension of the Cartan powers of the adjoint representation of a complex simple Lie algebra (i.e., a universal formula for the Hilbert functions of homogeneous complex contact manifolds), as well as several other universal formulas. These formulas generalize formulas of Vogel and Deligne and are given in terms of rational functions where both the numerator and denominator decompose into products of linear factors with integer coefficients. We discuss consequences of the formulas including a relation with Scorza varieties.     

Biderivations of finite-dimensional complex simple Lie algebras

In this paper, we prove that a biderivation of a finite-dimensional complex simple Lie algebra without the restriction of being skewsymmetric is an inner biderivation. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also obtain the forms of the linear commuting maps on the finite-dimensional complex simple Lie algebra or general linear Lie algebra.     

Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra g, the standard universal solution R(x)∈Uq(g)^⊗2 of the Quantum Dynamical Yang-Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang-Baxter Equation. It can be built from the standard R-matrix and from the solution F(x)∈Uq(g)^⊗2 of the Quantum Dynamical coCycle Equation as . F(x) can be computed explicitly as an infinite product through the use of an auxiliary linear equation, the ABRR equation.Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g=sl(n+1)(n?1) only, there could exist an element M(x)∈Uq(sl(n+1)) such that the dynamical gauge transform RJ of R(x) by M(x),

RJ=M₁⁻¹(x)M₂(xqh₁⁾⁻¹R(x)M₁(xqh₂)M₂(x),     

Infinitesimal deformations of restricted simple Lie algebras I

We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartan-type: the Witt–Jacobson and the Special Lie algebras.     

Gradings of positive rank on simple Lie algebras

We complete the classification of positive rank gradings on Lie algebras of simple algebraic groups over an algebraically closed field k whose characteristic is zero or not too small, and we determine the little Weyl groups in each case. We also classify the stable gradings and prove Popov’s conjecture on the existence of a Kostant section.