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For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{\"u}ller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic $G$-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of connected components of moduli spaces of such objects. We further exhibit an alternative explanation of fundamental results on counting components in the absence of a parabolic structure., Comment: 48 pages, 3 tables

A G-grading on an algebra, where G is an abelian group, is called multiplicity-free if each homogeneous component of the grading is 1-dimensional. We introduce skew root systems of Lie type and skew root systems of Jordan type, and use them to construct multiplicity-free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp., Jordan) algebras are simple. Two families of skew root systems of Lie type (resp., of Jordan type) are constructed and the corresponding Lie (resp., Jordan) algebras are identified. This is a new approach to study abelian group gradings on Lie and Jordan algebras.

We count the geodesics of a given length connecting two points of a compact connected Lie group with a biinvariant metric. We reduce the question to the maximal torus by using the lattice, the diagram and the Weyl group to count the geodesics that occur outside the maximal torus. We apply our results to give short proofs of known results on conjugate and cut points of compact semisimple Lie groups.

We continue investigations begun in our previous works where we proved that the phase diagram of the Toda system on special linear groups can be identified with the Bruhat order on the symmetric group if all eigenvalues of the Lax matrix are distinct or with the Bruhat order on permutations of a multiset if there are multiple eigenvalues. We show that the phase portrait of the Toda system and the Hasse diagram of the Bruhat order coincide in the case of an arbitrary simple Lie group of rank 2. For this, we verify this property for the two remaining rank-2 groups, Sp(4,ℝ) and the real form of G2.

Let G be a group and let K be a field of characteristic p>0. Lie nilpotent group algebras of strong Lie nilpotency index up to 11 have already been classified. In this paper, our aim is to classify...

Let K be a field of characteristic p>0 and let KG be the group algebra of an arbitrary group G over K. It is known that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is at least p+1. The group algebras KG for which these indices are p+1 or 2p or 3p−1 or 4p−2 have already been determined. In this paper, we classify the group algebras KG for which the upper Lie nilpotency index is 5p−3, 6p−4 or 7p−5.

Let ${\mathcal{A}}$ be a unital torsion-free algebra over a unital commutative ring ${\mathcal{R}}$. To characterise Lie $n$-higher derivations on ${\mathcal{A}}$, we give an identity which enables us to transfer problems related to Lie $n$-higher derivations into the same problems concerning Lie $n$-derivations. We prove that: (1) if every Lie $n$-derivation on ${\mathcal{A}}$ is standard, then so is every Lie $n$-higher derivation on ${\mathcal{A}}$; (2) if every linear mapping Lie $n$-derivable at several points is a Lie $n$-derivation, then so is every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points; (3) if every linear mapping Lie $n$-derivable at several points is a sum of a derivation and a linear mapping vanishing on all $(n-1)$th commutators of these points, then every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points is a sum of a higher derivation and a sequence of linear mappings vanishing on all $(n-1)$th commutators of these points. We also give several applications of these results.

We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ⊂ SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.

We prove that simple Lie pencils of rank 1 over an algebraically closed field P of characteristic 0 with operators of left multiplication being derivations are of the form of a sandwich algebra M 3(U,D′), where U is the subspace of all skew-symmetric matrices in M 3(P) and D′ is any subspace containing 〈E〉 in the space of all diagonal matrices D in M 3(P).

Einstein-like examples of four-dimensional Lorentzian Lie groups are listed and geometric properties of each class have been investigated.

We consider left-invariant f-structures on 6-dimensional filiform Lie groups and present sufficient conditions under which special left-invariant f-structures on these Lie groups belong to the class of Hermitian f-structures. We also give particular examples of the corresponding f-structures.

A flat Lorentzian Lie algebra is a left symmetric algebra endowed with a symmetric bilinear form of signature (−, +,…, +) such that left multiplications are skew-symmetric. In geometrical terms, a flat Lorentzian Lie algebra is the Lie algebra of a Lie group with a left-invariant Lorentzian metric with vanishing curvature. In this article, we show that any flat nonunimodular Lorentzian Lie algebras can be obtained as a double extension of flat Riemannian Lie algebras. As an application, we give all flat nonunimodular Lorentzian Lie algebras up to dimension 4.

In terms of sandwich algebras, we obtain a classification of symmetrical simple Lie sheaves over an algebraically closed field of zero characteristic.

We study compact homogeneous spaces of reductive Lie groups, and also some of their analogues and generalizations (quasicompact and plesiocompact homogeneous spaces of these Lie groups). We give a description of the structure of (plesio-)uniform subgroups in reductive Lie groups. The corresponding homogeneous spaces for which the stationary subgroup has an extremal dimension (close to the minimal or maximal possible one) are described. The fundamental groups of (plesio)compact homogeneous spaces of arbitrary reductive and semisimple Lie groups are characterized. Bibliography: 16 titles.

In this paper, we investigate an interplay between the hexagonal network models and the root system of a particular class of rank two Lie algebras, called $$G_2$$G2. In this work, we show that the hexagonal cells explored in telecommunication systems are associated with the nonzero roots of $$G_2$$G2-generalized Lie algebras. More precisely, the $$G_2$$G2 hexagons are analyzed in some details and they are shown to be linked to the equation defining the co-channel reuse ratio. Using root systems and Dynkin diagrams technics, we reveal that such a equation can be converted into an algebraic relation in terms of the two simple roots describing $$G_2$$G2-generalized Lie algebras.

We give a new version of the Gromov’s centralizer theorem in the case of semisimple Lie group actions and arbitrary rigid geometric structures of algebraic type. UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de investigaciones Matemáticas y Metamatemáticas (CIMM)

We consider spherical principal series representations of the semisimple Lie group of rank one $G=SO(n, 1; \mathbb K)$, $\mathbb K=\br, \bc, \bh$. There is a family of unitarizable representations $\pi_{\nu}$ of $G$ for $\nu$ in an interval on $\mathbb R^+$, the so-called complementary series, and subquotient or subrepresentations of $G$ for $\nu$ being negative integers. We consider the restriction of $(\pi_{\nu}, G)$ under the subgroup $H=SO(n-1, 1; \mathbb K)$. We prove the appearing of discrete components. The corresponding results for the exceptional Lie group $F_{4(-20)}$ and its subgroup $Spin(8,1)$ are also obtained., Comment: Revised version

We show that untwisted respectively twisted conjugacy classes of a compact and simply connected Lie group which satisfy a certain integrality condition correspond naturally to irreducible highest weight representations of the corresponding affine Lie algebra. Along the way, we review the classification of twisted conjugacy classes of a simply connected compact Lie group G and give a description of their stabilizers in terms of the Dynkin diagram of the corresponding twisted affine Lie algebra

This chapter is on the structure theory of complex, semisimple Lie algebras. We give complete details for \( \mathfrak{s}{{\mathfrak{l}}_{n}} \)(ℂ) and give a brief account for other semisimple Lie algebras. For details, see [26].

We classify the 6-dimensional Lie algebras of the form g × g that admit an integrable complex structure. We also endow a Lie algebra of the kind o ( n ) × o ( n ) ( n ≥ 2 ) with such a complex structure. The motivation comes from geometric structures a la Sasaki on g -manifolds.

In this chapter, we discuss the representation theory of complex semisimple Lie algebras. While we give full details for \( \mathfrak{s}{{\mathfrak{l}}_{n}} \)(ℂ), we only sketch the details for other semisimple Lie algebras. For more details, we refer the reader to [26].

Sharp forms of Kunze–Stein phenomena on S L ( 2 , R ) are obtained by using symmetrization and Stein–Weiss potentials. A new structural proof with remarkable simplicity can be given on S L ( 2 , R ) which effectively transfers the analysis from the group to the symmetric space corresponding to a manifold with negative curvature. Our methods extend to include the Lorentz groups and n -dimensional hyperbolic space through application of the Riesz–Sobolev rearrangement inequality. A new framework is developed for Riesz potentials on semisimple symmetric spaces and the semi-direct product of groups analogous to the Iwasawa decomposition for semisimple Lie groups. Extensions to higher-rank Lie groups and analysis on multidimensional connected hyperboloids including anti de Sitter space are suggested by the analysis outlined here.

Symmetry groups of Lie algebras and superalgebras constructed from (∈, δ)-Freudenthal-Kantor triple systems have been studied. In particular, for a special (ε, ε)-Freudenthal–Kantor triple, it is the SL(2) group. Also, the relationship between two constructions of Lie algebras from structurable algebras has been investigated.

The Killing form induces a 2-plectic structure on a compact semisimple Lie group. The associated Lie group of canonical transformations (2-plectomorphisms) is compact. This 2-plectic structure induces a Cartan connection on the Lie group. The curvature and torsion tensor of this connection have been calculated for the special unitary Lie group S U ( 3 ) . It is shown that the homogeneous spaces S U ( 3 ) S U ( 2 ) , S U ( 3 ) S 1 and S U ( 3 ) T 2 also, admit 2-plectic structures, which are induced by closed left invariant 3-forms on S U ( 3 ) , whereas S U ( 3 ) U ( 2 ) and S U ( 3 ) S O ( 3 ) do not admit such 2-plectic structures.

The authors consider a family of finite-dimensional Lie superalgebras of O-type over an algebraically closed field of characteristic p > 3. It is proved that the Lie superalgebras of O-type are simple and the spanning sets are determined. Then the spanning sets are employed to characterize the superderivation algebras of these Lie superalgebras. Finally, the associative forms are discussed and a comparison is made between these Lie superalgebras and other simple Lie superalgebras of Cartan type.

This paper shows that for any Lie group G whose Lie algebra L is the split real form of a complex simple Lie algebra, and for any arbitrary root ?, there exists a Cartan decomposition of L, related to?, which characterizes some controllability properties by using the adjoint orbits of sl(2,?)$\text {sl}(2, \mathbb {R})$. For a class of invariant control systems evolving on G, it is proved that the necessary full rank condition for controllability is also sufficient.

It is proved that all Morse height functions on regular orbits of the adjoint action of compact semisimple Lie groups are perfect. In the case of an arbitrary linear representation of a compact Lie group it is proved that all height functions satisfy the Bott property on the orbits of the representation. The case of the group SO4 is considered in more details.

In this paper we suggest an approach to the study of orbits of actions of semisimple Lie groups in their irreducible complex representations. This approach is based on differential invariants on the one hand, and on geometry of reductive homogeneous spaces on the other hand. According to Borel–Weil–Bott theorem, every irreducible representation of semisimple Lie group is isomorphic to the action of this group on the module of holomorphic sections of some one-dimensional bundle over homogeneous space. Using this, we give a complete description of the structure of the field of differential invariants for this action and obtain a criterion, which separates regular orbits.

A classical construction of Atiyah assigns to any (real or complex) Lie group G, manifold M and principal homogeneous G-space P over M, a Lie algebroid over M ([1]). The spirit behind our work is to put this work within an algebraic context, replace M by a scheme X and G by a “simple” reductive group scheme G over X (in the sense of Demazure–Grothendieck) that arise naturally with an attached torsor (which plays the role of P) in the study of Extended Affine Lie Algebras (see [9] for an overview). Lie algebroids in an algebraic sense were also considered by Beilinson and Bernstein. We will explain how the present work relates to theirs. Fil: Kuttler, Jochen. University of Alberta; Canadá Fil: Pianzola, Arturo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. University of Alberta; Canadá. Centro de Altos Estudios en Ciencias Exactas; Argentina Fil: Quallbrunn, Federico. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Centro de Altos Estudios en Ciencias Exactas; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Matemáticas; Argentina

For any abelian group $G$, we classify up to isomorphism all $G$-gradings on the classical central simple Lie algebras, except those of type $D_4$, over the field of real numbers (or any real closed field)., 35 pages

The main purpose in this paper is to completely determine the groups of isometries of simple para-Hermitian symmetric spaces. That enables us to also determine the groups of affine transformations of these spaces, with respect to the canonical affine connections.

We use the Killing form to derive the decomposition of a finite-dimensional semisimple Lie algebra over \(\mathbb {C}\) into a direct sum of simple ideals. Moreover, we prove the Weyl’s theorem of complete reducibility, and the equivalence of the complete reducibility of real and complex modules is also given. Cartan’s root-space decomposition of a finite-dimensional semisimple Lie algebra over \(\mathbb {C}\) is derived.

The aim of this note is to give a geometric proof for classical local rigidity of lattices in semisimple Lie groups. We are reproving well known results in a more geometric (and hopefully clearer) way., Comment: This is a 2004 paper uploaded for archiving purposes

The coadjoint representation of a connected algebraic group $Q$ with Lie algebra $\mathfrak q$ is a thrilling and fascinating object. Symmetric invariants of $\mathfrak q$ (= $\mathfrak q$-invariants in the symmetric algebra $S(\mathfrak q)$) can be considered as a first approximation to the understanding of the coadjoint action $(Q:\mathfrak q^*)$ and coadjoint orbits. In this article, we study a class of non-reductive Lie algebras, where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant-theoretic properties. If $G$ is a semisimple group with Lie algebra $\mathfrak g$ and $V$ is $G$-module, then we define $\mathfrak q$ to be the semi-direct product of $\mathfrak g$ and $V$. Then we are interested in the case, where the generic isotropy group for the $G$-action on $V$ is reductive and commutative. It turns out that in this case symmetric invariants of $\mathfrak q$ can be constructed via certain $G$-equivariant maps from $\mathfrak g$ to $V$ ("covariants")., Comment: 33 pages

Cyclic metric Lie groups are Lie groups equipped with a left-invariant metric which is in some way far from being biinvariant, in a sense made explicit in terms of Tricerri and Vanhecke’s homogeneous structures. The semisimple and solvable cases are studied. We extend to the general case, Kowalski–Tricerri’s and Bieszk’s classifications of connected and simply-connected unimodular cyclic metric Lie groups for dimensions less than or equal to five.

We study class of finite-dimensional Cantan-type Lie superalgebras HO(m) over a field of prime characteristic, which can be regarded as extensions of odd Hamiltonian superalgebra HO. And we also determine the derivation superalgebras of Lie superalgebras HO(m).

Let G be the complex connected simply connected simple Lie group of type G 2 or F 4 . Let K denote the fixed point subgroup relative to an involution of G that is lifted from a Cartan involution. This article gives a description of certain components of Springer fibers associated to closed K -orbits contained in the flag variety of G . These components allow us to describe certain multiplicity polynomials associated to discrete series representations of the real form G 2 2 of G 2 and the two real forms F 4 4 and F 4 − 20 of F 4 . The goals for this paper are motivated by the descriptions of Springer fiber components and the associated multiplicity polynomials for type S U ( p , q ) described in a paper of Barchini and Zierau.

This chapter deals with general theories that do not depend on the types of Lie groups and Lie algebras. As generalizations, it addresses projective representations of Lie groups and Lie algebras by combining the contents of Chap. 2. Then, it introduces the Fourier transform for Lie groups including the case of projective representations. It also prepares several concepts for Chap. 6. Also, this chapter introduces complex Lie groups and complex Lie algebras, which are helpful for real Lie groups and real Lie algebras.

Let G be a real Lie group, ? be a lattice in G and G be a compactly generated closed subgroup of G . If the Zariski closure of the group Ad(G) is semisimple with no compact factor, we prove that every G -orbit closure in G/? is a finite volume homogeneous space. We also establish related equidistribution properties.

We obtain a classification of 2and 3-dimensional Lie sheaves. DOI: 10.3103/S1066369X13110017

A Lie algebra g is called two step nilpotent if g is not abelian and [g, g] lies in the center of g. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension 8 over the field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension 8.

From a Lie algebra $\mathfrak{g}$ satisfying $\mathcal{Z}(\mathfrak{g})=0$ and $\Lambda^2(\mathfrak{g})^\mathfrak{g}=0$ (in particular, for $\g$ semisimple) we describe explicitly all Lie bialgebra structures on extensions of the form $\mathfrak{L} =\mathfrak{g}\times \mathbb{K}$ in terms of Lie bialgebra structures on $\mathfrak{g}$ (not necessarily factorizable nor quasi-triangular) and its biderivations, for any field $\mathbb{K}$ with char $\mathbb{K}=0$. If moreover, $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$, then we describe also all Lie bialgebra structures on extensions $\mathfrak{L} =\mathfrak{g}\times \mathbb{K}^n$. In interesting cases we characterize the Lie algebra of biderivations., Comment: 23 pages

We classify the full-rank left-invariant control affine systems evolving on (real) semisimple three-dimensional Lie groups. This is accomplished by reducing the problem to that of classifying the affine subspaces of the Lie algebras so (2; 1) and so (3).

We study left-invariant complex-valued harmonic morphisms from Riemannian Lie groups. We show that in each dimension greater than 3 there exist Riemannian Lie groups that do not have any such solutions.