Groups of automorphisms of Lie algebras such that the fixed-point subalgebra of each non-identity element is solvable

Abstract Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of $\textrm{ABCDEFG}$-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of $\textrm{ADE}$-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jacobian fibrations of Calabi–Yau orbifold stacks constructed by the 1st author. We construct simultaneous crepant resolutions of the associated singular quasi-projective Calabi–Yau three-folds and compare the resulting intermediate Jacobian fibrations to the corresponding Hitchin systems.

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Fixed-point-free automorphisms of Lie algebras

Acta Mathematica Sinica English Series ◽

10.1007/bf02107627 ◽

1989 ◽

Vol 5 (1) ◽

pp. 95-96 ◽

Cited By ~ 1

Author(s):

Zha Jianguo

Keyword(s):

Fixed Point ◽

Lie Algebras

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Fixed point subalgebras of extended affine Lie algebras

Journal of Algebra ◽

10.1016/j.jalgebra.2004.08.011 ◽

2005 ◽

Vol 287 (2) ◽

pp. 351-380 ◽

Cited By ~ 7

Author(s):

Saeid Azam ◽

Stephen Berman ◽

Malihe Yousofzadeh

Keyword(s):

Fixed Point ◽

Lie Algebras ◽

Affine Lie Algebras

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On groups of automorphisms of Lie algebras

Journal of Algebra ◽

10.1016/0021-8693(68)90038-0 ◽

1968 ◽

Vol 8 (2) ◽

pp. 131-142 ◽

Cited By ~ 19

Author(s):

David J Winter

Keyword(s):

Lie Algebras ◽

Groups Of Automorphisms

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Distribution Algebras on p-adic Groups and Lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-025-3 ◽

2011 ◽

Vol 63 (5) ◽

pp. 1137-1160 ◽

Cited By ~ 2

Author(s):

Allen Moy

Keyword(s):

Lie Algebras ◽

Identity Element ◽

Hecke Algebra ◽

Symmetric Bilinear Form ◽

Convolution Product ◽

Complex Vector ◽

Enveloping Algebra ◽

Invariant Distributions ◽

Reductive Lie Algebra ◽

The Family

Abstract When F is a p-adic field, and is the group of F-rational points of a connected algebraic F-group, the complex vector space of compactly supported locally constant distributions on G has a natural convolution product that makes it into a ℂ-algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for p-adic groups of the enveloping algebra of a Lie group. However, has drawbacks such as the lack of an identity element, and the process is not a functor. Bernstein introduced an enlargement . The algebra consists of the distributions that are left essentially compact. We show that the process is a functor. If is a morphism of p-adic groups, let be the morphism of ℂ-algebras. We identify the kernel of in terms of Ker. In the setting of p-adic Lie algebras, with g a reductive Lie algebra, m a Levi, and the natural projection, we show that maps G-invariant distributions on to NG(m)-invariant distributions on m. Finally, we exhibit a natural family of G-invariant essentially compact distributions on g associated with a G-invariant non-degenerate symmetric bilinear form on g and in the case of SL(2) show how certain members of the family can be moved to the group.

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ON AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE UNIVERSAL ALGEBRA

International Journal of Algebra and Computation ◽

10.1142/s0218196707003974 ◽

2007 ◽

Vol 17 (05n06) ◽

pp. 1085-1106 ◽

Cited By ~ 5

Author(s):

G. MASHEVITZKY ◽

B. I. PLOTKIN

Keyword(s):

Lie Algebras ◽

Universal Algebra ◽

Automorphism Groups ◽

Endomorphism Semigroup ◽

Universal Algebras ◽

Free Lie Algebras ◽

Inner Automorphism ◽

Groups Of Automorphisms ◽

Inner Automorphisms

Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

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Semiregular Automorphisms in Vertex-Transitive Graphs of Order $3p^2$

The Electronic Journal of Combinatorics ◽

10.37236/7499 ◽

2018 ◽

Vol 25 (2) ◽

Cited By ~ 1

Author(s):

Dragan Marušič

Keyword(s):

Fixed Point ◽

Prime Order ◽

Automorphism Groups ◽

Identity Element ◽

Permutation Groups ◽

Vertex Transitive ◽

Vertex Transitive Graphs ◽

Transitive Graphs ◽

Free Element

It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally $2$-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. It is the purpose of this paper to prove that vertex-transitive graphs of order $3p^2$, where $p$ is a prime, contain semiregular automorphisms.

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Some Remarks on Groups Admitting a Fixed-Point-Free Automorphism

Canadian Journal of Mathematics ◽

10.4153/cjm-1968-128-5 ◽

1968 ◽

Vol 20 ◽

pp. 1300-1307 ◽

Cited By ~ 8

Author(s):

Fletcher Gross

Keyword(s):

Fixed Point ◽

Finite Group ◽

Simple Group ◽

Direct Product ◽

Finite Simple Group ◽

Identity Element ◽

Simple Groups ◽

Cyclic Groups ◽

A Finite Group ◽

Open Question

A finite group G is said to be a fixed-point-free-group (an FPF-group) if there exists an automorphism a which fixes only the identity element of G. The principal open question in connection with these groups is whether non-solvable FPF-groups exist. One of the results of the present paper is that if a Sylow p-group of the FPF-group G is the direct product of any number of mutually non-isomorphic cyclic groups, then G has a normal p-complement. As a consequence of this, the conjecture that all FPF-groups are solvable would be true if it were true that every finite simple group has a non-trivial SylowT subgroup of the kind just described. Here it should be noted that all the known simple groups satisfy this property.

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