Real Representations of $C_2$-Graded Groups: The Linear and Hermitian Theories

Rumynin, Dmitriy; Taylor, J.R.

doi:10.48550/arxiv.2008.07846

Cited by 1 publication

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“…If G is connected, this is [7, Lemma 2.5]. In general, any connected component of G/H has a form G 1 gH/H for some g ∈ G. Since π 0 (G) = π 0 (G), there exists x ∈ H ∩ Gg so that G 1 gH/H = G 1 xH/H = G 1 /(xHx −1 ) and we can use [7,Lemma 2.5] to settle the general case.…”

Section: Bijection Since τ (Xhxmentioning

confidence: 99%

Compact Lie Groups and Complex Reductive Groups

Jones¹,

Rumynin²,

Childress³

2021

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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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Section: Bijection Since τ (Xhxmentioning

confidence: 99%