Triple automorphisms of simple Lie algebras

Deng Yin Wang; Xiaoxiang Yu

Displaying similar documents to “Triple automorphisms of simple Lie algebras”

The contact system on the $(m, ℓ)$ -jet spaces

J. Muñoz, F. J. Muriel, Josemar Rodríguez (2001)

Archivum Mathematicum

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This paper is a continuation of [MMR:98], where we give a construction of the canonical Pfaff system $Ω (M_{m}^{ℓ})$ on the space of $(m, ℓ)$ -velocities of a smooth manifold $M$ . Here we show that the characteristic system of $Ω (M_{m}^{ℓ})$ agrees with the Lie algebra of $Aut (ℝ_{m}^{ℓ})$ , the structure group of the principal fibre bundle ${\overset{ˇ}{M}}_{m}^{ℓ} ⟶ J_{m}^{ℓ} (M)$ , hence it is projectable to an irreducible contact system on the space of $(m, ℓ)$ -jets ( $= ℓ$ -th order contact elements of dimension $m$ ) of $M$ . Furthermore, we translate to the language of Weil bundles the structure...

The groups of automorphisms of the Witt $W_{n}$ and Virasoro Lie algebras

Vladimir V. Bavula (2016)

Czechoslovak Mathematical Journal

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Let $L_{n} = K [x_{1}^{\pm 1}, ..., x_{n}^{\pm 1}]$ be a Laurent polynomial algebra over a field $K$ of characteristic zero, $W_{n} : = {Der}_{K} (L_{n})$ the Lie algebra of $K$ -derivations of the algebra $L_{n}$ , the so-called Witt Lie algebra, and let $Vir$ be the Virasoro Lie algebra which is a $1$ -dimensional central extension of the Witt Lie algebra. The Lie algebras $W_{n}$ and $Vir$ are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: ${Aut}_{Lie} (Vir) ≃ {Aut}_{Lie} (W_{1}) ≃ {\pm 1} ⋉ K^{*}$ , and give a short proof that ${Aut}_{Lie} (W_{n}) ≃ {Aut}_{K - alg} (L_{n}) ≃ {GL}_{n} (ℤ) ⋉ K^{* n}$ .

Automorphisms of central extensions of type I von Neumann algebras

Sergio Albeverio, Shavkat Ayupov, Karimbergen Kudaybergenov, Rauaj Djumamuratov (2011)

Studia Mathematica

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Given a von Neumann algebra M we consider its central extension E(M). For type I von Neumann algebras, E(M) coincides with the algebra LS(M) of all locally measurable operators affiliated with M. In this case we show that an arbitrary automorphism T of E(M) can be decomposed as $T = T_{a} \circ T_{ϕ}$ , where $T_{a} (x) = a x a^{- 1}$ is an inner automorphism implemented by an element a ∈ E(M), and $T_{ϕ}$ is a special automorphism generated by an automorphism ϕ of the center of E(M). In particular if M is of type $I_{\infty}$ then every band preserving...

Algorithmic computations of Lie algebras cohomologies

Šilhan, Josef

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From the text: The aim of this work is to advertise an algorithmic treatment of the computation of the cohomologies of semisimple Lie algebras. The base is Kostant’s result which describes the representation of the proper reductive subalgebra on the cohomologies space. We show how to (algorithmically) compute the highest weights of irreducible components of this representation using the Dynkin diagrams. The software package $L i E$ offers the data structures and corresponding procedures for...

Degree sequences of graphs containing a cycle with prescribed length

Jian Hua Yin (2009)

Czechoslovak Mathematical Journal

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Let $r \geq 3$ , $n \geq r$ and $π = (d_{1}, d_{2}, ..., d_{n})$ be a non-increasing sequence of nonnegative integers. If $π$ has a realization $G$ with vertex set $V (G) = {v_{1}, v_{2}, ..., v_{n}}$ such that $d_{G} (v_{i}) = d_{i}$ for $i = 1, 2, ..., n$ and $v_{1} v_{2} \dots v_{r} v_{1}$ is a cycle of length $r$ in $G$ , then $π$ is said to be potentially $C_{r}^{''}$ -graphic. In this paper, we give a characterization for $π$ to be potentially $C_{r}^{''}$ -graphic.

Automorphisms of $(λ) / ℐ_{κ}$

Paul Larson, Paul McKenney (2016)

Fundamenta Mathematicae

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We study conditions on automorphisms of Boolean algebras of the form $(λ) / ℐ_{κ}$ (where λ is an uncountable cardinal and $ℐ_{κ}$ is the ideal of sets of cardinality less than κ ) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every automorphism of $(2^{κ}) / ℐ_{κ ⁺}$ which is trivial on all sets of cardinality κ⁺ is trivial, and that $M A_{ℵ ₁}$ implies both that every automorphism of (ℝ)/Fin is trivial on a cocountable set and that every automorphism of (ℝ)/Ctble is trivial. ...

The Banach-Lie Group of Lie Automorphisms of an $H^{*}$ -Algebra

Antonio J. Calderón Martín, Candido Martín González (2007)

Bollettino dell'Unione Matematica Italiana

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We study the Banach-Lie group $\operatorname{Aut}(A^{-})$ of Lie automorphisms of a complex associative $H^{*}$ -algebra. Also some consequences about its Lie algebra, the algebra of Lie derivations of $A$ , are obtained. For a topologically simple $A$ , in the infinite-dimensional case we have $\operatorname{Aut}(A^{-})_{0}=\operatorname{Aut}(A)$ implying $\operatorname{Der}(A)=\operatorname{Der}(A^{-})$ . In the finite dimensional case $\operatorname{Aut}(A^{-})_{0}$ is a direct product of $\operatorname{Aut}(A)$ and a certain subgroup of Lie derivations $\delta$ from $A$ to its center, annihilating commutators.

The Strong Anick Conjecture is true

Vesselin Drensky, Jie-Tai Yu (2007)

Journal of the European Mathematical Society

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Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra $K 〈 x, y, z 〉$ over a field $K$ of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of $K 〈 x, y, z 〉$ . In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We...

Automorphisms of Spacetime Manifold with Torsion

Vladimir Ivanovich Pan’Zhenskii, Olga Petrovna Surina (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper we prove that the maximum dimension of the Lie group of automorphisms of the Riemann–Cartan 4-dimensional manifold does not exceed 8, and if the Cartan connection is skew-symmetric or semisymmetric, the maximum dimension is equal to 7. In addition, in the case of the Riemann–Cartan $n$ -dimensional manifolds with semisymmetric connection the maximum dimension of the Lie group of automorphisms is equal to $n (n - 1) / 2 + 1$ for any $n > 2$ .

Multiplicative maps that are close to an automorphism on algebras of linear transformations

L. W. Marcoux, H. Radjavi, A. R. Sourour (2013)

Studia Mathematica

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Let be a complex, separable Hilbert space of finite or infinite dimension, and let ℬ() be the algebra of all bounded operators on . It is shown that if φ: ℬ() → ℬ() is a multiplicative map(not assumed linear) and if φ is sufficiently close to a linear automorphism of ℬ() in some uniform sense, then it is actually an automorphism; as such, there is an invertible operator S in ℬ() such that $φ (A) = S^{- 1} A S$ for all A in ℬ(). When is finite-dimensional, similar results are obtained with the mere assumption...