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 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) .

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.

On p-local homotopy types of gauge groups

2014 ◽  
Vol 144 (1) ◽  
pp. 149-160 ◽  
Author(s):  
Daisuke Kishimoto ◽  
Akira Kono ◽  
Mitsunobu Tsutaya
Keyword(s):  
Gauge Group ◽  
Homotopy Type ◽  
Principal Bundle ◽  
Dimensional Sphere ◽  
Gauge Groups ◽  
Homotopy Types

The aim of this paper is to show that the p-local homotopy type of the gauge group of a principal bundle over an even-dimensional sphere is completely determined by the divisibility of the classifying map by p. In particular, for gauge groups of principal SU(n)-bundles over S2d for 2 ≤ d ≤ p − 1 and n ≤ 2p − 1, we give a concrete classification of their p-local homotopy types.

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).

Homotopy commutativity in p-localized gauge groups

2013 ◽  
Vol 143 (4) ◽  
pp. 851-870 ◽  
Author(s):  
D. Kishimoto ◽  
A. Kono ◽  
S. Theriault
Keyword(s):  
Gauge Group ◽  
Lie Group ◽  
Chern Class ◽  
Compact Lie Group ◽  
Gauge Groups

Let G be a simple, compact Lie group and let $\mathcal{G}_k(G)$ be the gauge group of the principal G-bundle over S4 with second Chern class k. McGibbon classified the groups G that are homotopy commutative when localized at a prime p. We show that in many cases the homotopy commutativity of G, or its failure, determines that of $\mathcal{G}_k(G)$.

Unstable K1-group and homotopy type of certain gauge groups

2006 ◽  
Vol 136 (1) ◽  
pp. 149-155 ◽  
Author(s):  
Hiroaki Hamanaka ◽  
Akira Kono
Keyword(s):  
Gauge Group ◽  
Homotopy Type ◽  
Complete Classification ◽  
Necessary Condition ◽  
Gauge Groups ◽  
Homotopy Types

We denote the group of homotopy set [X, U(n)] by the unstable K1-group of X. In this paper, using the unstable K1-group of the multi-suspended CP2, we give a necessary condition for two principal SU(n)-bundles over §4 to have the associated gauge group of the same homotopy type, which is an improvement of the result of Sutherland and, particularly, show the complete classification of homotopy types of SU(3)-gauge groups over S4.

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