Complex abelian Lie groups with finite-dimensional cohomology groups

AbstractI consider a two-parameter family Bs,t of unitary transforms mapping an L2-space over a Lie group of compact type onto a holomorphic L2-space over the complexified group. These were studied using infinite-dimensional analysis in joint work with B. Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generalizations of the classical Segal-Bargmann transform. I consider also the limiting cases s → ∞ and s → t/2.

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Cylindrical representations of some infinite dimensional nuclear Lie groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700011916 ◽

1992 ◽

Vol 46 (2) ◽

pp. 295-310 ◽

Cited By ~ 1

Author(s):

Jean Marion

Keyword(s):

Lie Groups ◽

Lie Group ◽

Additive Group ◽

Test Functions ◽

Necessary And Sufficient Condition ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Irreducible Unitary Representations ◽

Direct Product Group ◽

Necessary And Sufficient

Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie groupGand for any strictly positive integerkthere exist non located and non trivially decomposable representations of orderkof the nuclear Lie group(M;G) of all theG-valued test-functions on a given paracompact manifoldM.

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TOWARD SYMMETRIC SPACES OF AFFINE KAC–MOODY TYPE

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887806001648 ◽

2006 ◽

Vol 03 (05n06) ◽

pp. 881-898 ◽

Cited By ~ 4

Author(s):

ERNST HEINTZE

Keyword(s):

Fixed Point ◽

Lie Groups ◽

Symmetric Spaces ◽

Point Group ◽

Infinite Dimensional ◽

Simple Lie Groups ◽

Finite Dimensional ◽

Expository Article

In this expository article we discuss some ideas and results which might lead to a theory of infinite dimensional symmetric spaces [Formula: see text] where [Formula: see text] is an affine Kac–Moody group and [Formula: see text] the fixed point group of an involution (of the second kind). We point out several striking similarities of these spaces with their finite dimensional counterparts and discuss their geometry. Furthermore we sketch a classification and show that they are essentially in 1 : 1 correspondence with hyperpolar actions on compact simple Lie groups.

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INTRODUCTION TO QUANTIZED LIE GROUPS AND ALGEBRAS

International Journal of Modern Physics A ◽

10.1142/s0217751x92002805 ◽

1992 ◽

Vol 07 (25) ◽

pp. 6175-6213 ◽

Cited By ~ 30

Author(s):

T. TJIN

Keyword(s):

Lie Groups ◽

Quantum Groups ◽

Hopf Algebras ◽

Baxter Equation ◽

Product Decomposition ◽

R Matrix ◽

Conformal Field ◽

Finite Dimensional ◽

Lie Groups And Algebras ◽

Poisson Lie Groups

We give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups we study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of sl2 is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal R matrix for the quantum sl2 algebra. In the last section we deduce all finite-dimensional irreducible representations for q a root of unity. We also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.

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An Open Mapping Theorem For Pro-Lie Groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700036387 ◽

2007 ◽

Vol 83 (1) ◽

pp. 55-78 ◽

Cited By ~ 2

Author(s):

Karl H. Hofmann ◽

Sidney A. Morris

Keyword(s):

Lie Groups ◽

Projective Limit ◽

Factor Group ◽

Continuous Group ◽

Closed Graph ◽

Graph Theorem ◽

Closed Graph Theorem ◽

Open Mapping Theorem ◽

Finite Dimensional ◽

Group Modulo

AbstractA pro-Lie group is a projective limit of finite dimensional Lie groups. It is proved that a surjective continuous group homomorphism between connected pro-Lie groups is open. In fact this remains true for almost connected pro-Lie groups where a topological group is called almost connected if the factor group modulo the identity component is compact. As consequences we get a Closed Graph Theorem and the validity of the Second Isomorphism Theorem for pro-Lie groups in the almost connected context.

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Exotic Holonomies ${\rm E}^{(a)}_7$

International Journal of Mathematics ◽

10.1142/s0129167x97000305 ◽

1997 ◽

Vol 08 (05) ◽

pp. 583-594 ◽

Cited By ~ 6

Author(s):

Quo-Shin Chi ◽

Sergey Merkulov ◽

Lorenz Schwachhöfer

Keyword(s):

Symmetry Group ◽

Lie Groups ◽

Lie Group ◽

Moduli Spaces ◽

Local Symmetry ◽

Positive Dimension ◽

Affine Connections ◽

Finite Dimensional ◽

Torsion Free

It is proved that the Lie groups [Formula: see text] and [Formula: see text] represented in ℝ56 and the Lie group [Formula: see text] represented in ℝ112 occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension.

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