The length and depth of compact Lie groups

Burness, Timothy C.; Liebeck, Martin W.; Shalev, Aner

doi:10.1007/s00209-019-02324-7

Cited by 6 publications

(5 citation statements)

References 21 publications

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“…We prove the following result, which refines and extends [3,Lemma 2.1] to not necessarily connected subgroups. Theorem 0.2.…”

supporting

confidence: 60%

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Compact Lie groups and complex reductive groups

Jones,

Rumynin,

Thomas

2024

Homology, Homotopy and Applications

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“…We prove the following result, which refines and extends [3,Lemma 2.1] to not necessarily connected subgroups. Theorem 0.2.…”

supporting

confidence: 60%

“…In the opposite direction, every ϕ ∈ hom(A(G), C) yields a collection (x i ) of linear operators V i → V i by (3). This extends to an element x = (x V ) ∈ T (G) by using the canonical decompositions using the coefficient vector spaces hom(V i , V )…”

Section: Properties Of Complexificationmentioning

confidence: 99%

Compact Lie groups and complex reductive groups

Jones,

Rumynin,

Thomas

2024

Homology, Homotopy and Applications

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“…We prove the following result, which refines and extends [2,Lemma 2.1] to not necessarily connected subgroups.…”

supporting

confidence: 60%

Compact Lie Groups and Complex Reductive Groups

Jones¹,

Rumynin²,

Childress³

2021

Preprint

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We show that the categories of compact Lie groups and complex reductive groups are equivalent as infinity categories. The groups are not assumed to be connected.In this note we examine the categories Lie = {Compact Lie Groups} and Red = {Complex Reductive Groups}, where the morphisms are group homomorphisms, preserving the additional structures. The categories Lie and Red are not equivalent. Although the isomorphism classes are in one-toone correspondence, the morphisms are not: just compare the automorphisms of SU 2 and SL 2 (C).The aim of this note is to fix the aforementioned lack of equivalence by proving the following theorem.Theorem 1. There exists a complexification functor T : Lie −→ Red with the following four properties.(1) The group G ∈ Lie is a maximal compact subgroup of T (G).(2) The functor T is injective on morphisms.(3) The functor T is essentially surjective on objects. (4) For all H, G ∈ Lie the embedding T H,G : hom(H, G) → hom(H, G) is a homotopy equivalence. Here the hom-sets are equipped with the compact open topology. Theorem 1 admits an interpretation (cf. [7, 1.1]) in a naive version of ∞-categories, outlined by Lurie [6, Def. 1.1.1.6]. Let Top be the closed monoidal category of compactly generated, weakly Hausdorff topological spaces. Both Lie and Red are categories, enriched in Top: take the Lie group topology on the groups and the compact-open topology on the hom-sets. Given X ∈ Top we denote its weak homotopy type by [[X]] ∈ Ho(Top). Given a category C, enriched in Top, we denote its homotopy category, enriched in Ho(Top), by [[C]]. This category has the same objects as C and [[C]](X, Y ) := [[C(X, Y )]]. Theorem 1 essentially establishes an Ho(Top)-enriched equivalence [[T ]] of the enriched categories [[Lie]] and [[Red]]. Alternatively, Theorem 1 yields an ∞-equivalence of the ∞-categories [[Lie]] and [[Red]]. Our secondary aim is to investigate how the complexification functor behaves on subgroups. Suppose G = T (G). Consider the sets S(G) := {compact subgroups of G}, S(G) := {reductive subgroups of G}.

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“…Different aspects of word maps are considered in a vast and extended literature; we refer to the papers [7,8,9,20,21,45,46,47,57,72,97,98,99,101] for details, surveys and further explanations. Waring type questions for rings were considered by Matei Brešar [17].…”

Section: Evaluations Of Wordsmentioning

confidence: 99%

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Kanel-Belov¹,

Malev

Rowen

et al. 2020

SIGMA

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Let p be a polynomial in several non-commuting variables with coefficients in a field K of arbitrary characteristic. It has been conjectured that for any n, for p multilinear, the image of p evaluated on the set M n (K) of n by n matrices is either zero, or the set of scalar matrices, or the set sl n (K) of matrices of trace 0, or all of M n (K). This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for n = 2 in Section 2, some decisive results for n = 3 in Section 3, and partial information for n ≥ 3 in Section 4, also for non-multilinear polynomials. In addition we consider the case of K not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other researches, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of art, and in the other hand providing counterexamples showing the boundaries of generalizations.

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The length and depth of compact Lie groups

Cited by 6 publications

References 21 publications

Compact Lie groups and complex reductive groups

Compact Lie groups and complex reductive groups

Compact Lie Groups and Complex Reductive Groups

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

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