Simple Lie group
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Survey on lie group machine learning Open
Lie group machine learning is recognized as the theoretical basis of brain intelligence, brain learning, higher machine learning, and higher artificial intelligence. Sample sets of Lie group matrices are widely available in practical appli…
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Expansion in finite simple groups of Lie type Open
We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].
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On low-dimensional solvmanifolds Open
A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups w…
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Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations Open
We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectoria…
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Strong property (T) for higher-rank simple Lie groups: Figure 1. Open
We prove that connected higher rank simple Lie groups have Lafforgue's strong\nproperty (T) with respect to a certain class of Banach spaces\n$\\mathcal{E}_{10}$ containing many classical superreflexive spaces and some\nnon-reflexive space…
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Connectedness of Higgs bundle moduli for complex reductive Lie groups Open
We carry an intrinsic approach to the study of the connectedness of the moduli space $\mathcal{M}_G$ of $G$-Higgs bundles, over a compact Riemann surface, when $G$ is a complex reductive (not necessarily connected) Lie group. We prove that…
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Compact Lie groups: Euler constructions and generalized Dyson conjecture Open
A generalized Euler parameterization of a compact Lie group is a way for parameterizing the group starting from a maximal Lie subgroup, which allows a simple characterization of the range of parameters. In the present paper we consider the…
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Einstein metrics on compact simple Lie groups attached to standard triples Open
In this paper, we study left invariant Einstein metrics on compact simple Lie groups. We find a method to construct left invariant non-naturally reductive Einstein metrics on compact simple Lie groups from a standard triple. This result, c…
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Group actions on dendrites and curves Open
We establish obstructions for groups to act by homeomorphisms on dendrites. For instance, lattices in higher rank simple Lie groups will always fix a point or a pair. The same holds for irreducible lattices in products of connected groups.…
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Geometrical aspects of the Lie algebra S-expansion procedure Open
In this article it is shown that S-expansion procedure affects the geometry of a Lie group, changing it and leading us to the geometry of another Lie group with higher dimensionality. A method for determining the semigroup, which would pro…
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Matrix 3-Lie superalgebras and BRST supersymmetry Open
Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if in…
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The Isometry Group of an RCD ∗ Space is Lie Open
We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, w…
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Lie algebraic Carroll/Galilei duality Open
We characterize Lie groups with bi-invariant bargmannian, galilean, or carrollian structures. Localizing at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian, or galilean structures are actually determined b…
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Alternating subgroups of exceptional groups of Lie type Open
In this paper we examine embeddings of alternating groups and symmetric\ngroups into almost simple groups of exceptional type. In particular, we prove\nthat unless the alternating or symmetric group has degree 6 or 7, there is no\nmaximal …
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Expanding Cone and Applications to Homogeneous Dynamics Open
Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calc…
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Graphs and two-step nilpotent Lie algebras Open
We consider a method popular in the literature of associating a two-step nilpotent Lie algebra with a finite simple graph. We prove that the two-step nilpotent Lie algebras associated with two graphs are Lie isomorphic if and only if the g…
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Automorphisms of fusion systems of finite simple groups of Lie type Open
For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic, wi…
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Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups Open
Let G be a connected complex simple Lie group, and let G ^ d {\widehat{G}^{\mathrm{d}}} be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that G ^ d {\wide…
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Measurable regularity properties of infinite-dimensional Lie groups Open
We consider differential equations of the form y'(t)=f(t,y(t)) on a (possibly infinite-dimensional) Lie group G, for f : [0,1] x G -> TG a time-dependent left invariant vector field with measurable (but not necessarily continuous) dependen…
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Diophantine properties of nilpotent Lie groups Open
A finitely generated subgroup ${\rm\Gamma}$ of a real Lie group $G$ is said to be Diophantine if there is ${\it\beta}>0$ such that non-trivial elements in the word ball $B_{{\rm\Gamma}}(n)$ centered at $1\in {\rm\Gamma}$ never approach the…
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Real semisimple Lie groups and balanced metrics Open
Given any non-compact real simple Lie group \mathrm{G}_o of inner type and even dimension, we prove the existence of an invariant complex structure \mathrm{J} and a Hermitian balanced metric on \mathrm{G}_o and on any compact quotient \mat…
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Projective unitary representations of infinite-dimensional Lie groups Open
For an infinite-dimensional Lie group $G$ modeled on a locally convex Lie algebra ${\\mathfrak{g}}$ , we prove that every smooth projective unitary representation of $G$ corresponds to a smooth linear unitary representation of a Lie group …
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Pro-Lie Groups: A Survey with Open Problems Open
A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and fo…
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Indefinite Einstein metrics on nice Lie groups Open
We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat …
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On Some Lie Groups Containing Spin Group in Clifford Algebra Open
In this paper we consider some Lie groups in complexified Clifford algebras.\nUsing relations between operations of conjugation in Clifford algebras and\nmatrix operations we prove isomorphisms between these groups and classical\nmatrix gr…
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Integrating central extensions of Lie algebras via Lie 2-groups Open
The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of L…
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Almost Lie Algebroids and Characteristic Classes Open
Almost Lie algebroids are generalizations of Lie algebroids, when the\nJacobiator is not necessary null. A simple example is given, for which a Lie\nalgebroid bracket or a Courant bundle is not possible for the given anchor, but\na natural…
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Continuous Groups for Physicists Open
Continuous Groups for Physicists is written for graduate students as well as researchers working in the field of theoretical physics. The text has been designed uniquely and it balances coverage of advanced and non-standard topics with an …
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On Lie-central extensions of Leibniz algebras Open
Basing ourselves on the categorical notions of central extensions and commutators in the framework of semi-abelian categories relative to a Birkhoff subcategory, we study central extensions of Leibniz algebras with respect to the Birkhoff …
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Lie groupoids of mappings taking values in a Lie groupoid Open
Endowing differentiable functions from a compact manifold to a Lie group with\nthe pointwise group operations one obtains the so-called current groups and, as\na special case, loop groups. These are prime examples of infinite-dimensional\n…