Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

On the volume of orbifold quotients of symmetric spaces

Key to H. C. Wang's quantitative study of Zassenhaus neighbourhoods of non-compact semisimple Lie groups are two constants that depend on the root system of the corresponding Lie algebra. This article extends the list of values for Wang's constants to the exceptional Lie groups and also removes their dependence on dimension. The first application is an improved upper sectional curvature bound for a canonical left-invariant metric on a semisimple Lie group. The second application is an explicit uniform positive lower bound for arbitrary orbifold quotients of a given irreducible symmetric space of non-compact type.     

On the factor sets of measures and local tightness of convolution semigroups over Lie groups

It is shown that for a large class of Lie groups (called weakly algebraic groups) including all connected semisimple Lie groups the following holds: for any probability measure on the Lie group the set of all two-sided convolution factors is compact if and only if the centralizer of the support of inG is compact. This is applied to prove that for any connected Lie groupG, any homomorphism of any real directed (submonogeneous) semigroup into the topological semigroup of all probability measures onG is locally tight.     

Note on Co-split Lie Algebras

It is proved that,any finite dimensional complex Lie algebra L = [L,L],hence,over a field of characteristic zero,any finite dimensional Lie algebra L = [L,L] possessing a basis with complex structure constants,should be a weak co-split Lie algebra.A class of non-semi-simple co-split Lie algebras is given.     

Critical Landscape Topology for Optimization on the Symplectic Group

Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over a noncompact symplectic Lie group Sp(2N,ℝ), i.e., minimizing the Frobenius distance from a target symplectic transformation, which can be used to assess the fidelity function over dynamical transformations in classical mechanics and quantum optics. The topology of the set of critical points is proven to have a unique local minimum and a number of saddlepoint submanifolds, exhibiting the absence of local suboptima that may hinder the search for ultimate optimal solutions. Compared with those of previously studied problems on compact Lie groups, such as the orthogonal and unitary groups, the topology is more complicated due to the significant nonlinearity brought by the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group.     

Cocycles Over Abelian TNS Actions

We study extensions of higher-rank Abelian TNS actions (i.e. hyperbolic and with a special structure of the stable distributions) by compact connected Lie groups. We show that up to a constant, there are only finitely many cohomology classes. We also show the existence of cocycles over higher-rank Abelian TNS actions that are not cohomologous to constant cocycles. This is in contrast to earlier results, showing that real valued cocycles, or small Lie group valued cocycles, over higher-rank Abelian actions are cohomologous to constants.     

Homogeneous Ricci solitons on four-dimensional Lie groups with a left-invariant Riemannian metric

Homogeneous Ricci solitons on four-dimensional Lie groups with a left-invariant Riemannian metric are studied. The absence of nontrivial homogeneous invariant Ricci solitons is proved. The algebraic soliton equations are solved in terms of the structure constants of the metric Lie algebra.     

Different definitions of homogeneity of real hypersurfaces in ℂ2

The coincidence of two definitions of local homogeneity for real-analytic hypersurfaces in two-dimensional complex spaces is proved. It is shown that if any two germs of a Levi nondegenerate nonspherical surfaceM are equivalent, then this surface has a local Lie group structure:M then acts transitively on itself by left shifts, and each such shift is a local holomorphic transformation of ².Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 881–887, December, 1998.This paper was conceived and largely written during author's visits to the Universities of West Ontario and Regina (Canada) under the kind invitation of Professors B. Gilligan and A. Bowen. The author wishes to express his great indebtedness to them for providing excellent conditions for work and gratitude for useful discussions.     

Rozansky–Witten invariants via Atiyah classes

Recently, L. Rozansky and E. Witten associated to any hyper-Kähler manifold X a system of weights (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to the existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyper-Kähler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work at the level of cohomology and for any Kähler manifold, if we work at the level of Dolbeault cochains. As an outcome of our considerations, we give a formula for Rozansky–Witten classes using any Kähler metric on a holomorphic symplectic manifold.     

On balanced Hermitian structures on Lie groups

We present several methods for the construction of balanced Hermitian structures on Lie groups. In our methods a partial differential equation is involved so that the resulting structures are in general non homogeneous. In particular, we prove that for 3-step nilpotent Lie groups G of dimension 6, any left-invariant complex structure on G admits a balanced Hermitian metric. Starting from normal almost contact structures, we construct balanced metrics on 6-dimensional manifolds, generalizing warped products. Finally, explicit balanced Hermitian structures are also given on solvable Lie groups defined as semidirect products ${\mathbb{R}^k \ltimes \mathbb{R}^{2n-k}}$ .     

Essential conformal actions of $${{\mathrm{PSL}}}(2,\mathbf {R})$$ on real-analytic compact Lorentz manifolds

The main result of this paper is the conformal flatness of real-analytic compact Lorentz manifolds of dimension at least three admitting a conformal essential action of a Lie group locally isomorphic to \({{\mathrm{PSL}}}(2,\mathbf {R})\). It is established by using a general result on local isometries of real-analytic rigid geometric structures. As corollary, we deduce the same conclusion for conformal essential actions of connected semi-simple Lie groups on real-analytic compact Lorentz manifolds. This work is a contribution to the understanding of the Lorentzian version of a question asked by A. Lichnerowicz.     

THE LIE ALGEBRA ADMITTED BY ANORDINARY DIFFERENTIAL EQUATION SYSTEM

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Local Superderivatio ns o n Lie Superalgebra $$\mathfrak{}$$ ( n)

Let \(\mathfrak{q}\)(n) be a simple strange Lie superalgebra over the complex field ?. In a paper by A.Ayupov, K.Kudaybergenov (2016), the authors studied the local derivations on semi-simple Lie algebras over ? and showed the difference between the properties of local derivations on semi-simple and nilpotent Lie algebras. We know that Lie superalgebras are a generalization of Lie algebras and the properties of some Lie superalgebras are similar to those of semi-simple Lie algebras, but \(\mathfrak{p}\)(n) is an exception. In this paper, we introduce the definition of the local superderivation on \(\mathfrak{q}\)(n), give the structures and properties of the local superderivations of \(\mathfrak{q}\)(n), and prove that every local superderivation on \(\mathfrak{q}\)(n), n > 3, is a superderivation.     

Higher-Dimensional Automorphic Lie Algebras

The paper presents the complete classification of Automorphic Lie Algebras based on \({{\mathfrak {sl}}}_{n}(\mathbb {C})\), where the symmetry group G is finite and acts on \({{\mathfrak {sl}}}_n(\mathbb {C})\) by inner automorphisms, \({{\mathfrak {sl}}}_n(\mathbb {C})\) has no trivial summands, and where the poles are in any of the exceptional G-orbits in \(\overline{\mathbb {C}}\). A key feature of the classification is the study of the algebras in the context of classical invariant theory. This provides on the one hand a powerful tool from the computational point of view; on the other, it opens new questions from an algebraic perspective (e.g. structure theory), which suggest further applications of these algebras, beyond the context of integrable systems. In particular, the research shows that this class of Automorphic Lie Algebras associated with the \(\mathbb {T}\mathbb {O}\mathbb {Y}\) groups (tetrahedral, octahedral and icosahedral groups) depend on the group through the automorphic functions only; thus, they are group independent as Lie algebras. This can be established by defining a Chevalley normal form for these algebras, generalising this classical notion to the case of Lie algebras over a polynomial ring.     

On characteristic operator-functions of Lie-algebras

An algebra Lie of linear operators in a Hilbert space with given characteristic function and given structure constants is defined up to a unitary transformation of the Hilbert space.This paper was originally published in Russian in "Proc. of Kharkov University", U.S.S.R. Kharkov 1972, No. 83, pp.41–45.The editor is grateful to S. Levin (Ben-Gurion University of the Negev) for the translation.     

Special symplectic Lie groups and hypersymplectic Lie groups

A special symplectic Lie group is a triple ${(G,\omega,\nabla)}$ such that G is a finite-dimensional real Lie group and ω is a left invariant symplectic form on G which is parallel with respect to a left invariant affine structure ${\nabla}$ . In this paper starting from a special symplectic Lie group we show how to “deform” the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure ${\nabla}$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.     

Error diffusion on acute simplices: invariant tiles

Using the notion of compatibility between Poisson brackets and cluster structures in the coordinate rings of simple Lie groups, Gekhtman, Shapiro and Vainshtein conjectured the existence of a cluster structure for each Belavin-Drinfeld solution of the classical Yang-Baxter equation compatible with the corresponding Poisson-Lie bracket on the simple Lie group. Poisson-Lie groups are classified by the Belavin-Drinfeld classification of solutions to the classical Yang-Baxter equation. For any non-trivial Belavin-Drinfeld data of minimal size for SL _n , the companion paper constructed a cluster structure with a locally regular initial seed, which was proved to be compatible with the Poisson bracket associated with that Belavin-Drinfeld data.This paper proves the rest of the conjecture: the corresponding upper cluster algebra \(\overline {{A_\mathbb{C}}} \left( C \right)\) is naturally isomorphie to O (SL _n ), the torus determined by the BD triple generates the action of \({\left( {\mathbb{C}*} \right)^{2{k_T}}}\) on C (SL _n ), and the correspondence between Belavin-Drinfeld classes and cluster structures is one to one.     

Classification of complex simply connected homogeneous spaces of dimension not greater than 2

The paper is devoted to the classification of finite-dimensional complex Lie algebras of analytic vector fields on the complex plane and the corresponding actions of Lie groups on complex two-dimensional manifolds. These Lie algebras were specified by Sophus Lie. He specified vector fields which form bases of the Lie algebras. However the structure of the Lie algebras was not clarified, and isomorphic Lie algebras among listed were not established. Thus, the classification was far from complete, and the situation has not been essentially changed until now. This paper is devoted to the completion of the above mentioned classification. We consider the part of this classification which concerns transitive actions of Lie groups.     

Group theoretical aspects of constants of motion and separable solutions in classical mechanics

The aim of this paper is to establish the group nature of all separable solutions and conserved quantities in classical mechanics by analyzing the group structure of the Hamilton-Jacobi equation. It is shown that consideration of only classical Lie point groups is insufficient. For this purpose the Lie-Bäcklund groups of tangent transformations, rigorously established by Ibragimov and Anderson, are used. It is also shown how these generalized groups induce Lie groups on Hamilton's equations. The generalization of the above results to any order partial differential equation, where the dependent variable does not appear explicitly, is obvious. In the second part of the paper we investigate a certain class of admissible operators of the time-independent Hamilton-Jacobi equation of any energy state including the zero state. It is shown that in the latter case additional symmetries may appear. Finally, some potentials of physical interest admitting higher symmetries are considered.