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Tree algebras: An algebraic axiomatization of intertwining vertex operators

Igor Kříž, Yang Xiu (2012)

Archivum Mathematicum

We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a tree algebra. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over ℂ . We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over ℚ .

Triple automorphisms of simple Lie algebras

Deng Yin Wang, Xiaoxiang Yu (2011)

Czechoslovak Mathematical Journal

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 .

Triple derivations on von Neumann algebras

Robert Pluta, Bernard Russo (2015)

Studia Mathematica

It is well known that every derivation of a von Neumann algebra into itself is an inner derivation and that every derivation of a von Neumann algebra into its predual is inner. It is less well known that every triple derivation (defined below) of a von Neumann algebra into itself is an inner triple derivation. We examine to what extent all triple derivations of a von Neumann algebra into its predual are inner. This rarely happens but it comes close. We prove a (triple) cohomological...

Troisième théorème fondamental de réalisation de Cartan

Ngô van Quê, A.A.M. Rodrigues (1975)

Annales de l'institut Fourier

De même qu’avec les groupes de Lie, à tout pseudo-groupe infinitésimal de Lie θ sur R n il est associé de façon naturelle une algèbre de Lie L ( θ ) , qui est une sous-algèbre de Lie fermée de l’algèbre de Lie D de tous les champs de vecteurs formels de R n , l’algèbre D étant munie de la topologie définie par la filtration naturelle de l’algèbre des séries formelles. Le troisième théorème fondamental de Cartan dit qu’inversement étant donnée une sous-algèbre de Lie transitive fermée L de l’algèbre D , il existe...

Two remarks on Lie rings of 2 × 2 matrices over commutative associative rings

Evgenii L. Bashkirov (2020)

Commentationes Mathematicae Universitatis Carolinae

Let C be an associative commutative ring with 1. If a ∈ C , then a C denotes the principal ideal generated by a . Let l , m , n be nonzero elements of C such that m n ∈ l C . The set of matrices a 11 a 12 a 21 - a 11 , where a 11 ∈ l C , a 12 ∈ m C , a 21 ∈ n C , forms a Lie ring under Lie multiplication and matrix addition. The paper studies properties of these Lie rings.

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