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

1997 ◽  
Vol 08 (05) ◽  
pp. 583-594 ◽  
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.

Some Results in the Connective K-Theory of Lie Groups

1988 ◽  
Vol 31 (2) ◽  
pp. 194-199
Author(s):  
L. Magalhães
Keyword(s):  
Hopf Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
K Theory ◽  
Torsion Free ◽  
Connected Lie Group

AbstractIn this paper we give a description of:(1) the Hopf algebra structure of k*(G; L) when G is a compact, connected Lie group and L is a ring of type Q(P) so that H*(G; L) is torsion free;(2) the algebra structure of k*(G2; L) for L = Z2 or Z.

A New Form of the Segal-Bargmann Transform for Lie Groups of Compact Type

1999 ◽  
Vol 51 (4) ◽  
pp. 816-834 ◽  
Author(s):  
Brian C. Hall
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Joint Work ◽  
Compact Type ◽  
Bargmann Transform ◽  
Infinite Dimensional ◽  
Infinite Dimensional Analysis ◽  
Finite Dimensional ◽  
New Form ◽  
Two Parameter

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.

scholarly journals Cylindrical representations of some infinite dimensional nuclear Lie groups

1992 ◽  
Vol 46 (2) ◽  
pp. 295-310 ◽  
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.

scholarly journals Invariant Poisson–Nijenhuis structures on Lie groups and classification

2018 ◽  
Vol 15 (04) ◽  
pp. 1850059 ◽  
Author(s):  
Zohreh Ravanpak ◽  
Adel Rezaei-Aghdam ◽  
Ghorbanali Haghighatdoost
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Symmetry Group ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Text Structure ◽  
Baxter Equation ◽  
Text Structures ◽  
Group A

We study right-invariant (respectively, left-invariant) Poisson–Nijenhuis structures ([Formula: see text]-[Formula: see text]) on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra [Formula: see text]. We show that [Formula: see text]-[Formula: see text] structures can be used to find compatible solutions of the classical Yang–Baxter equation (CYBE). Conversely, two compatible [Formula: see text]-matrices from which one is invertible determine an [Formula: see text]-[Formula: see text] structure. We classify, up to a natural equivalence, all [Formula: see text]-matrices and all [Formula: see text]-[Formula: see text] structures with invertible [Formula: see text] on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by [Formula: see text]-matrices on a phase space whose symmetry group is Lie group a [Formula: see text], can be specifically determined.

scholarly journals Pro-Lie groups which are infinite-dimensional Lie groups

2009 ◽  
Vol 146 (2) ◽  
pp. 351-378 ◽  
Author(s):  
K. H. HOFMANN ◽  
K.-H. NEEB
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Exponential Function ◽  
Lie Group ◽  
Smooth Manifold ◽  
Projective Limit ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Infinite Dimensional Lie Groups

AbstractA pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

scholarly journals Fixed points of Lie group actions on surfaces

1986 ◽  
Vol 6 (1) ◽  
pp. 149-161 ◽  
Author(s):  
J. F. Plante
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lie Groups ◽  
Euler Characteristic ◽  
Lie Group ◽  
Stationary Point ◽  
Compact Surface ◽  
The Other ◽  
Continuous Action ◽  
Finite Dimensional

AbstractLetGbe a connected finite-dimensional Lie group andMa compact surface. We investigate whether, for a givenGandM, every continuous action ofGonMmust have a fixed (stationary) point. It is shown that whenGis nilpotent andMhas non-zero Euler characteristic that every action ofGonMmust have a fixed point. On the other hand, it is shown that the non-abelian 2-dimensional Lie group (affine group of the line) acts without fixed points on every compact surface. These results make it possible to complete this investigation for Lie groups of dimension at most 3.

scholarly journals Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup

2009 ◽  
Vol 74 (3) ◽  
pp. 891-900 ◽  
Author(s):  
Alessandro Berarducci
Keyword(s):  
Compact Group ◽  
Recent Work ◽  
Lie Groups ◽  
Lie Group ◽  
Compact Groups ◽  
Compact Lie Group ◽  
Minimal Structure ◽  
Divisible Subgroup ◽  
Torsion Free ◽  
Cohomology Of Groups

AbstractBy recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.

scholarly journals Topological duality in humanoid robot dynamics

2001 ◽  
Vol 43 (2) ◽  
pp. 183-194 ◽  
Author(s):  
V. Ivancevic ◽  
C. E. M. Pearce
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Humanoid Robot ◽  
Robot Dynamics ◽  
Robot System ◽  
Topological Structures ◽  
Topological Duality ◽  
Finite Dimensional

AbstractA humanoid robot system may be viewed as a collection of segments coupled at rotational joints which geometrically represent constrained rotational Lie groups. This allows a study of the dynamics of the motion of a humanoid robot. Several formulations are possible. In this paper, dual invariant topological structures are constructed and analyzed on the finite-dimensional manifolds associated with the humanoid motion. Both cohomology and homology structures are examined on the tangent (Lagrangian) as well as on the cotangent (Hamiltonian) bundles on the manifold of the humanoid motion configuration. represented by the toral Lie group. It is established all four topological structures give in essence the same description of humanoid dynamics. Practically this means that whichever of these approaches we use, ultimately we obtain the same mathematical results.

Construction of Levi processes on path spaces of Lie groups

2016 ◽  
Vol 19 (01) ◽  
pp. 1650002
Author(s):  
A. A. Kalinichenko
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Weak Limit ◽  
Convolution Semigroup ◽  
Path Space ◽  
Probability Measures ◽  
Compact Lie Group ◽  
Piecewise Constant ◽  
Finite Dimensional

Given a compact Lie group and a conjugate-invariant Levi process on it, generated by the operator [Formula: see text], we construct the Levi process on the path space of [Formula: see text], associated with the convolution semigroup [Formula: see text] of probability measures, where [Formula: see text] is the distribution of the Levi process on [Formula: see text] generated by [Formula: see text]. The constructed process is obtained as the weak limit of piecewise constant paths, which, as well as proving its existence and properties, provides finite-dimensional approximations of Chernoff type to the integrals with respect to its distribution.

Feynman approximation to integrals with respect to Brownian sheet on Lie groups

2015 ◽  
Vol 18 (01) ◽  
pp. 1550008 ◽  
Author(s):  
A. A. Kalinichenko
Keyword(s):  
Brownian Motion ◽  
Lie Groups ◽  
Lie Group ◽  
Functional Space ◽  
Brownian Sheet ◽  
Functional Integrals ◽  
Finite Dimensional ◽  
Approximation Formulas ◽  
The One ◽  
Connected Lie Group

We consider the Feynman-type approximations to functional integrals over the distribution of the Brownian sheet on a compact connected Lie group M, which give a representation of the integrals over the functional space C([0, 1] × [0, 1], M) as the limit of integrals over the finite-dimensional manifolds M × ⋯ × M. The known approximation formulas for the one-parameter Brownian motion are generalized to the case of the Brownian sheet.

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