On the β-Construction in K-Theory

1972 ◽  
Vol 24 (5) ◽  
pp. 819-824
Author(s):  
C. M. Naylor
Keyword(s):  
Lie Group ◽  
Chern Character ◽  
Complex Representation ◽  
Group Homomorphism ◽  
Unique Element ◽  
Compact Lie Group ◽  
Fundamental Representation ◽  
K Theory ◽  
Simply Connected

The β-construction assigns to each complex representation φ of the compact Lie group G a unique element β(φ) in (G). For the details of this construction the reader is referred to [1] or [5]. The purpose of the present paper is to determine some of the properties of the element β(φ) in terms of the invariants of the representation φ. More precisely, we consider the following question. Let G be a simple, simply-connected compact Lie group and let f : S3 →G be a Lie group homomorphism. Then (S3) ⋍ Z with generator x = β(φ1), φ1 the fundamental representation of S3 , so that if φ is a representation of G,f*(φ) = n(φ)x, where n(φ) is an integer depending on φ and f . The problem is to determine n(φ).Since G is simple and simply-connected we may assume that ch2, the component of the Chern character in dimension 4 takes its values in H4(SG,Z)≅Z. Let u be a generator of H4(SG,Z) so that ch2(β (φ)) = m(φ)u, m(φ) an integer depending on φ.

scholarly journals On the K-theory of the loop space of a Lie group

1974 ◽  
Vol 76 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Francis Clarke
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Simple Structure ◽  
Homology Theory ◽  
Compact Lie Group ◽  
Classifying Space ◽  
K Theory ◽  
Simply Connected ◽  
Torsion Free ◽  
Multiplicative Structures

Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).

scholarly journals Unitary spherical representations of Drinfeld doubles

2018 ◽  
Vol 2018 (742) ◽  
pp. 157-186 ◽  
Author(s):  
Yuki Arano
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Complete Classification ◽  
Unitary Representations ◽  
Compact Lie Group ◽  
Drinfeld Double ◽  
Quantum Analogue ◽  
Property T ◽  
Simply Connected

Abstract We study irreducible spherical unitary representations of the Drinfeld double of the q-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of \mathrm{SU}_{q}(3) , we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group \mathrm{SU}_{q}(2n+1) has property (T), which also implies central property (T) of the dual of \mathrm{SU}_{q}(2n+1) .

scholarly journals Equivariant K -theory of compact Lie group actions with maximal rank isotropy

2012 ◽  
Vol 5 (2) ◽  
pp. 431-457 ◽  
Author(s):  
Alejandro Adem ◽  
José Manuel Gómez
Keyword(s):  
Lie Group ◽  
Group Actions ◽  
Maximal Rank ◽  
Compact Lie Group ◽  
Lie Group Actions ◽  
K Theory

scholarly journals THE FUNDAMENTAL GROUP OF G-MANIFOLDS

2013 ◽  
Vol 15 (03) ◽  
pp. 1250056 ◽  
Author(s):  
HUI LI
Keyword(s):  
Fixed Point ◽  
Fundamental Group ◽  
Lie Group ◽  
Maximal Torus ◽  
Moment Map ◽  
Compact Lie Group ◽  
Identity Component ◽  
Connected Subgroup ◽  
Stabilizer Group ◽  
Simply Connected

Let G be a connected compact Lie group, and let M be a connected Hamiltonian G-manifold with equivariant moment map ϕ. We prove that if there is a simply connected orbit G ⋅ x, then π1(M) ≅ π1(M/G); if additionally ϕ is proper, then π1(M) ≅ π1 (ϕ-1(G⋅a)), where a = ϕ(x). We also prove that if a maximal torus of G has a fixed point x, then π1(M) ≅ π1(M/K), where K is any connected subgroup of G; if additionally ϕ is proper, then π1(M) ≅ π1(ϕ-1(G⋅a)) ≅ π1(ϕ-1(a)), where a = ϕ(x). Furthermore, we prove that if ϕ is proper, then [Formula: see text] for all a ∈ ϕ(M), where [Formula: see text] is any connected subgroup of G which contains the identity component of each stabilizer group; in particular, π1(M/G) ≅ π1(ϕ-1(G⋅a)/G) for all a ∈ ϕ(M).

scholarly journals Equivariant K-theory of compact Lie groups with involution

2014 ◽  
Vol 13 (2) ◽  
pp. 313-335
Author(s):  
Po Hu ◽  
Igor Kriz ◽  
Petr Somberg
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Compact Lie Groups ◽  
K Theory ◽  
Simply Connected

AbstractFor a compact simply connected simple Lie group G with an involution α, we compute the G ⋊ ℤ/2-equivariant K-theory of G where G acts by conjugation and ℤ/2 acts either by α or by g ↦ α(g)−1. We also give a representation-theoretic interpretation of those groups, as well as of KG(G).

K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism

2001 ◽  
Vol 53 (3) ◽  
pp. 631-672 ◽  
Author(s):  
Samuel G. Walters
Keyword(s):  
Chern Character ◽  
Dimensional Subspace ◽  
Group Homomorphism ◽  
Computational Tool ◽  
Slight Generalization ◽  
K Theory ◽  
Continuous Fields

AbstractFor a dense Gδ set of real parameters θ in [0, 1] (containing the rationals) it is shown that the group K0(Aθ ⋊σ) is isomorphic to , where Aθ is the rotation C*-algebra generated by unitaries U, V satisfying VU = e2πiθUV and σ is the Fourier automorphism of Aθ defined by σ(U) = V, σ(V) = U−1. More precisely, an explicit basis for K0 consisting of nine canonical modules is given. (A slight generalization of this result is also obtained for certain separable continuous fields of unital C*-algebras over [0, 1].) The Connes Chern character ch: K0(Aθ ⋊σ) → Hev (Aθ ⋊σ)* is shown to be injective for a dense Gδ set of parameters θ. The main computational tool in this paper is a group homomorphism T: K0(Aθ ⋊σ) → obtained from the Connes Chern character by restricting the functionals in its codomain to a certain nine-dimensional subspace of Hev (Aθ ⋊σ). The range of T is fully determined for each θ. (We conjecture that this subspace is all of Hev.)

scholarly journals HOMOTOPY TYPES OF GAUGE GROUPS OVER NON-SIMPLYCONNECTED CLOSED 4-MANIFOLDS

2018 ◽  
Vol 61 (2) ◽  
pp. 349-371 ◽  
Author(s):  
TSELEUNG SO
Keyword(s):  
Homotopy Type ◽  
Lie Group ◽  
Compact Lie Group ◽  
Gauge Groups ◽  
Homotopy Types ◽  
Simply Connected

AbstractLet G be a simple, simply connected, compact Lie group, and let M be an orientable, smooth, connected, closed 4-manifold. In this paper, we calculate the homotopy type of the suspension of M and the homotopy types of the gauge groups of principal G-bundles over M when π1(M) is (1) ℤ*m, (2) ℤ/prℤ, or (3) ℤ*m*(*nj=1ℤ/prjjℤ), where p and the pj's are odd primes.

scholarly journals Twisted equivariant K-theory of compact Lie group actions with maximal rank isotropy

2018 ◽  
Vol 59 (11) ◽  
pp. 113502
Author(s):  
Alejandro Adem ◽  
José Cantarero ◽  
José Manuel Gómez
Keyword(s):  
Lie Group ◽  
Group Actions ◽  
Maximal Rank ◽  
Compact Lie Group ◽  
Lie Group Actions ◽  
K Theory

scholarly journals HIGHER GENERATION BY ABELIAN SUBGROUPS IN LIE GROUPS

2021 ◽  
Author(s):  
O. ANTOLÍN-CAMARENA ◽  
S. GRITSCHACHER ◽  
B. VILLARREAL
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Discrete Group ◽  
Short Note ◽  
Compact Lie Group ◽  
Abelian Subgroups ◽  
Simply Connected

AbstractTo a compact Lie group G one can associate a space E(2;G) akin to the poset of cosets of abelian subgroups of a discrete group. The space E(2;G) was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and Gómez, and other authors. In this short note, we prove that G is abelian if and only if πi(E(2;G)) = 0 for i = 1; 2; 4. This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply connected if and only if the group is abelian.

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