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Search: All articles in the CMB digital archive with keyword Lie group

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1. CMB Online first

Torres, David Martínez
The Diffeomorphism Type of Canonical Integrations Of Poisson tensors on Surfaces
A surface $\Sigma$ endowed with a Poisson tensor $\pi$ is known to admit canonical integration, $\mathcal{G}(\pi)$, which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\pi$ is not an area form on the 2-sphere, then $\mathcal{G}(\pi)$ is diffeomorphic to the cotangent bundle $T^*\Sigma$. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.

Keywords:Poisson tensor, Lie groupoid, cotangent bundle
Categories:58H05, 55R10, 53D17

2. CMB 2014 (vol 58 pp. 69)

Fulp, Ronald Owen
Correction to "Infinite Dimensional DeWitt Supergroups and Their Bodies"
The Theorem below is a correction to Theorem 3.5 in the article entitled " Infinite Dimensional DeWitt Supergroups and Their Bodies" published in Canad. Math. Bull. Vol. 57 (2) 2014 pp. 283-288. Only part (iii) of that Theorem requires correction. The proof of Theorem 3.5 in the original article failed to separate the proof of (ii) from the proof of (iii). The proof of (ii) is complete once it is established that $ad_a$ is quasi-nilpotent for each $a$ since it immediately follows that $K$ is quasi-nilpotent. The proof of (iii) is not complete in the original article. The revision appears as the proof of (iii) of the revised Theorem below.

Keywords:super groups, body of super groups, Banach Lie groups
Categories:58B25, 17B65, 81R10, 57P99

3. CMB 2012 (vol 57 pp. 132)

Mubeena, T.; Sankaran, P.
Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups
Given a group automorphism $\phi:\Gamma\longrightarrow \Gamma$, one has an action of $\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$. The orbits of this action are called $\phi$-twisted conjugacy classes. One says that $\Gamma$ has the $R_\infty$-property if there are infinitely many $\phi$-twisted conjugacy classes for every automorphism $\phi$ of $\Gamma$. In this paper we show that $\operatorname{SL}(n,\mathbb{Z})$ and its congruence subgroups have the $R_\infty$-property. Further we show that any (countable) abelian extension of $\Gamma$ has the $R_\infty$-property where $\Gamma$ is a torsion free non-elementary hyperbolic group, or $\operatorname{SL}(n,\mathbb{Z}), \operatorname{Sp}(2n,\mathbb{Z})$ or a principal congruence subgroup of $\operatorname{SL}(n,\mathbb{Z})$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.

Keywords:twisted conjugacy classes, hyperbolic groups, lattices in Lie groups

4. CMB 2011 (vol 55 pp. 870)

Wang, Hui; Deng, Shaoqiang
Left Invariant Einstein-Randers Metrics on Compact Lie Groups
In this paper we study left invariant Einstein-Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

Keywords:Einstein-Randers metric, compact Lie groups, geodesic, flag curvature
Categories:17B20, 22E46, 53C12

5. CMB 2004 (vol 47 pp. 119)

Theriault, Stephen D.
$2$-Primary Exponent Bounds for Lie Groups of Low Rank
Exponent information is proven about the Lie groups $SU(3)$, $SU(4)$, $Sp(2)$, and $G_2$ by showing some power of the $H$-space squaring map (on a suitably looped connected-cover) is null homotopic. The upper bounds obtained are $8$, $32$, $64$, and $2^8$ respectively. This null homotopy is best possible for $SU(3)$ given the number of loops, off by at most one power of~$2$ for $SU(4)$ and $Sp(2)$, and off by at most two powers of $2$ for $G_2$.

Keywords:Lie group, exponent

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