scholarly journals A criterion for discrete branching laws for Klein four symmetric pairs and its application to E6(−14)

2020 ◽  
Vol 31 (06) ◽  
pp. 2050049
Author(s):  
Haian He
Keyword(s):  
Lie Group ◽  
Simple Formula ◽  
Complete Classification ◽  
Necessary Condition ◽  
Symmetric Pair ◽  
Branching Laws ◽  
Nonidentity Element

Let [Formula: see text] be a noncompact connected simple Lie group, and [Formula: see text] a Klein four-symmetric pair. In this paper, we show a necessary condition for the discrete decomposability of unitarizable simple [Formula: see text]-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for [Formula: see text], there does not exist a unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. As an application, for [Formula: see text], we obtain a complete classification of Klein four symmetric pairs [Formula: see text], with [Formula: see text] noncompact, such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module and is also discretely decomposable as a [Formula: see text]-module for some nonidentity element [Formula: see text].

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 On the Balmer spectrum for compact Lie groups

2019 ◽  
Vol 156 (1) ◽  
pp. 39-76
Author(s):  
Tobias Barthel ◽  
J. P. C. Greenlees ◽  
Markus Hausmann
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Complete Classification ◽  
Prime Ideals ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Finite Set

We study the Balmer spectrum of the category of finite $G$-spectra for a compact Lie group $G$, extending the work for finite $G$ by Strickland, Balmer–Sanders, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of $G$. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes $p$.

scholarly journals CONVEXLY ORDERABLE GROUPS AND VALUED FIELDS

2014 ◽  
Vol 79 (01) ◽  
pp. 154-170
Author(s):  
JOSEPH FLENNER ◽  
VINCENT GUINGONA
Keyword(s):  
Abelian Groups ◽  
Complete Classification ◽  
Necessary Condition ◽  
Valued Fields ◽  
Ordered Groups ◽  
Algebraic Theories ◽  
Convexly Orderable

Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.

DISCRETE DECOMPOSABLE BRANCHING LAWS AND PROPER MOMENTUM MAPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250021 ◽  
Author(s):  
SALMA NASRIN
Keyword(s):  
Lie Group ◽  
Discrete Series ◽  
Geometric Quantization ◽  
Series Representation ◽  
Coadjoint Orbit ◽  
Irreducible Unitary Representation ◽  
Symmetric Pair ◽  
Holomorphic Discrete Series ◽  
Branching Laws ◽  
Scalar Type

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions. (1) The restriction π|H is discretely decomposable in the sense of Kobayashi. (2) The momentum map [Formula: see text] is proper. In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

On independent permutation separability criteria

2006 ◽  
Vol 6 (3) ◽  
pp. 277-288
Author(s):  
L. Clarisse ◽  
P. Wocjan
Keyword(s):  
Complete Classification ◽  
Necessary Condition ◽  
Numerical Verification ◽  
Separability Criteria

Recently, P.\ Wocjan and M.\ Horodecki [Open Syst.\ Inf.\ Dyn.\ 12, 331 (2005)] gave a characterization of combinatorially independent permutation separability criteria. Combinatorial independence is a necessary condition for permutations to yield truly independent criteria meaning that no criterion is strictly stronger that any other. In this paper we observe that some of these criteria are still dependent and analyze why these dependencies occur. To remove them we introduce an improved necessary condition and give a complete classification of the remaining permutations. We conjecture that the remaining class of criteria only contains truly independent permutation separability criteria. Our conjecture is based on the proof that for two, three and four parties all these criteria are truly independent and on numerical verification of their independence for up to 8 parties. It was commonly believed that for three parties there were 9 independent criteria, here we prove that there are exactly 6 independent criteria for three parties and 22 for four parties.

scholarly journals A Boothby–Wang Theorem for Besse Contact Manifolds

2020 ◽  
Author(s):  
Marc Kegel ◽  
Christian Lange
Keyword(s):  
Lie Group ◽  
Group Actions ◽  
Contact Manifold ◽  
Complete Classification ◽  
Contact Manifolds ◽  
Lie Group Actions ◽  
Oxford University ◽  
Reeb Flows ◽  
Sasakian Geometry

AbstractA Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal $$S^1$$ S 1 -orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition, this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry (Boyer and Galicki in Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008). We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way, we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner–Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism.

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.

Inequalities in Discrete Subgroups of PSL(2, R)

1988 ◽  
Vol 40 (1) ◽  
pp. 115-130 ◽  
Author(s):  
Jane Gilman
Keyword(s):  
Discrete Group ◽  
Sufficient Conditions ◽  
Complete Classification ◽  
Necessary Condition ◽  
Discrete Subgroups ◽  
Elementary Subgroup ◽  
Van Vleck ◽  
Necessary And Sufficient ◽  
Number Of Authors

Conditions for a subgroup, F, of PSL(2, R) to be discrete have been investigated by a number of authors. Jørgensen's inequality [5] gives an elegant necessary condition for discreteness for subgroups of PSL(2, C). Purzitsky, Rosenberger, Matelski, Knapp, and Van Vleck, among others [12, 13, 14, 9, 16, 17, 18, 19, 20, 7, 21] studied two generator discrete subgroups of PSL(2, R) in a long series of papers. The complete classification of two generator subgroups was surprisingly complicated and elusive. The most complete result appears in [20].In this paper we use the results of [20] to prove that a nonelementary subgroup F of PSL(2, R) is discrete if and only if every non-elementary subgroup, G, generated by two hyperbolics is discrete (Theorem 5.2) and that F contains no elliptics if and only if each such G is free (Theorem 5.1). Thus, we produce necessary and sufficient conditions for a non-elementary subgroup F of PSL(2, R) to be a discrete group without elliptic elements (Theorem 6.1) or a discrete group containing only hyperbolic elements (Theorem 7.1).

scholarly journals Discretely decomposable restrictions of (𝔤,K)-modules for Klein four symmetric pairs of exceptional Lie groups of Hermitian type

2019 ◽  
Vol 31 (01) ◽  
pp. 2050001 ◽  
Author(s):  
Haian He
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Simple Formula ◽  
Symmetric Pair ◽  
Exceptional Lie Groups

Let [Formula: see text] be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs [Formula: see text] such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module [Formula: see text] that is discretely decomposable as a [Formula: see text]-module. In this paper, three assumptions will be made. First, [Formula: see text] is an exceptional Lie group of Hermitian type, i.e. [Formula: see text] or [Formula: see text]. Second, [Formula: see text] is noncompact. Third, there exists an element [Formula: see text] corresponding to a symmetric pair of anti-holomorphic type such that [Formula: see text] is discretely decomposable as a [Formula: see text]-module.

scholarly journals Rigidity of maximal holomorphic representations of Kähler groups

2015 ◽  
Vol 26 (14) ◽  
pp. 1550113 ◽  
Author(s):  
Marco Spinaci
Keyword(s):  
Riemann Surface ◽  
Lie Group ◽  
Complete Classification ◽  
Schwarz Lemma ◽  
Connected Components ◽  
Equivariant Map ◽  
Kähler Groups ◽  
Diagonal Embedding

We investigate representations of Kähler groups [Formula: see text] to a semisimple non-compact Hermitian Lie group [Formula: see text] that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor–Wood inequality similar to those found by Burger–Iozzi and Koziarz–Maubon. Thanks to the study of the case of equality in Royden’s version of the Ahlfors–Schwarz lemma, we can completely describe the case of maximal holomorphic representations. If [Formula: see text], these appear if and only if [Formula: see text] is a ball quotient, and essentially reduce to the diagonal embedding [Formula: see text]. If [Formula: see text] is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, which thus appear as preferred elements of the respective maximal connected components.

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