Transformation Groups最新文献

We describe a straightforward construction of the pseudo-split absolutely pseudo-simple groups of minimal type with irreducible root systems of type (BC_n); these exist only in characteristic 2. We also give a formula for the dimensions of their irreducible modules.

We introduce the class of permawound unipotent groups, and show that they simultaneously satisfy certain “ubiquity” and “rigidity” properties that in combination render them very useful in the study of general wound unipotent groups. As an illustration of their utility, we present two applications: We prove that nonsplit smooth unipotent groups over (infinite) fields finitely generated over (textbf{F}_p) have infinite first cohomology; and we show that every commutative p-torsion wound unipotent group over a field of degree of imperfection 1 is the maximal unipotent quotient of a commutative pseudo-reductive group, thus partially answering a question of Totaro.

In this paper, we classify all Leavitt path algebras which have the property that every Lie ideal is an ideal. As an application, we show that Leavitt path algebras with this property provide a class of locally finite, infinite-dimensional Lie algebras whose locally solvable radical is completely determined. This particularly gives us a new class of semisimple Lie algebras over a field of prime characteristic.

The rationality problem of two-dimensional purely quasi-monomial actions was solved completely by (Hoshi, Kang and Kitayama, J. Algebra 403, 363-400, 2014). As a generalization, we solve the rationality problem of two-dimensional quasi-monomial actions under the condition that the actions are defined within the base field. In order to prove the theorem, we give a brief review of the Severi-Brauer variety with some examples and rationality results. We also use a rationality criterion for conic bundles of (mathbb {P}^1) over non-closed fields.

We provide a parameterization of all fusion subcategories of the equivariantization by a group action on a fusion category. As applications, we classify the Hopf subalgebras of a family of semisimple Hopf algebras of Kac-Paljutkin type and recover Naidu-Nikshych-Witherspoon classification of the fusion subcategories of the representation category of a twisted quantum double of a finite group.

Abstract

A pseudo-Riemannian Lie group ((G,langle cdot ,cdot rangle )) is a connected and simply connected Lie group with a left-invariant pseudo-Riemannian metric of signature (p, q). This paper is to study pseudo-Riemannian Lie group ((G,langle cdot ,cdot rangle )) with conformal vector fields induced by derivations. Firstly, we show that if (mathfrak {h}) is a Cartan subalgebra for a semisimple Levi factor of ({mathfrak g}) , where ({mathfrak g}) denotes the Lie algebra of G, then (dim mathfrak {h}le max {0,min {p,q}-1}) . It implies that ({mathfrak g}) is solvable for both Riemannian (i.e., (min {p,q}=0) ) and Lorentzian (i.e., (min {p,q}=1) ) cases, and furthermore we prove that (mathfrak {sl}_2(mathbb {R})) is the only possible Levi factor for the trans-Lorentzian (i.e., (min {p,q}=2) ) case. Secondly, based on the classification of the Riemannian and Lorentzian cases in (Corrigendum J. Algebra 603, 38–40 2022), we prove that the Riemannian Lie groups are of constant zero sectional curvature, hence conformally flat; for the Lorentzian case, we obtain a simple criterion for such Lorentzian Lie groups to be conformally flat, and moreover, we show that they are steady algebraic Ricci soliton with vanishing scalar curvature. Finally, we remark that the first known examples of homogeneous essential Lorentzian manifolds that are non-conformally flat (Translation in Siberian Math. J. 33, 1087–1093 1992), are isometric to Lorentzian Lie groups with conformal vector fields induced by derivations.

Abstract

We construct a new class of quantum vertex algebras associated with the normalized Yang R-matrix. They are obtained as Yangian deformations of certain (mathcal {S}) -commutative quantum vertex algebras, and their (mathcal {S}) -locality takes the form of a single RTT-relation. We establish some preliminary results on their representation theory and then further investigate their braiding map. In particular, we show that its fixed points are closely related with Bethe subalgebras in the Yangian quantization of the Poisson algebra (mathcal {O}(mathfrak {gl}_N((z^{-1})))) , which were recently introduced by Krylov and Rybnikov. Finally, we extend this construction of commutative families to the case of trigonometric R-matrix of type A.

In this paper, we study homogeneous sub-Riemannian manifolds whose normal extremals are the orbits of one-parameter subgroups of the group of smooth isometries (abbreviated as sub-Riemannian geodesic orbit manifolds). Following Tóth’s approach, we first obtain a sufficient and necessary condition for a homogeneous sub-Riemannian manifold to be geodesic orbit. Secondly, we study left-invariant sub-Riemannian geodesic orbit metrics on connected and simply connected nilpotent Lie groups. It turns out that every sub-Riemannian geodesic orbit nilmanifold is the restriction of a Riemannian geodesic orbit nilmanifold. Thirdly, we provide a method to construct compact and non-compact sub-Riemannian geodesic orbit manifolds and present a large number of sub-Riemannian geodesic orbit manifolds from Tamaru’s classification of Riemannian geodesic orbit manifolds fibered over irreducible symmetric spaces. Finally, we give a complete description of sub-Riemannian geodesic orbit metrics on spheres, and show that many of sub-Riemannian geodesic orbit manifolds have no abnormal sub-Riemannian geodesics.

Abstract

We characterize Riemannian orbifolds with an upper curvature bound in the Alexandrov sense as reflectofolds, i.e., Riemannian orbifolds all of whose local groups are generated by reflections, with the same upper bound on the sectional curvature. Combined with a result by Lytchak–Thorbergsson this implies that a quotient of a Riemannian manifold by a closed group of isometries has locally bounded curvature (from above) in the Alexandrov sense if and only if it is a reflectofold.

Golan and Sapir proved that Thompson’s groups F, T and V have linear divergence. In the current paper, we focus on the divergence property of several generalisations of the Thompson groups. We first consider the Brown-Thompson groups (F_n), (T_n) and (V_n) (also called Brown-Higman-Thompson group in some other context) and find that these groups also have linear divergence functions. We then focus on the braided Thompson groups BF, (widehat{BF}) and (widehat{BV}) and prove that these groups have linear divergence. The case of BV has also been done independently by Kodama.