• Title/Summary/Keyword: solvable Lie group

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NILPOTENCY OF THE RICCI OPERATOR OF PSEUDO-RIEMANNIAN SOLVMANIFOLDS

  • Huihui An;Shaoqiang Deng;Zaili Yan
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.3
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    • pp.867-873
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    • 2024
  • A pseudo-Riemannian solvmanifold is a solvable Lie group endowed with a left invariant pseudo-Riemannian metric. In this short note, we investigate the nilpotency of the Ricci operator of pseudo-Riemannian solvmanifolds. We focus on a special class of solvable Lie groups whose Lie algebras can be expressed as a one-dimensional extension of a nilpotent Lie algebra ℝD⋉n, where D is a derivation of n whose restriction to the center of n has at least one real eigenvalue. The main result asserts that every solvable Lie group belonging to this special class admits a left invariant pseudo-Riemannian metric with nilpotent Ricci operator. As an application, we obtain a complete classification of three-dimensional solvable Lie groups which admit a left invariant pseudo-Riemannian metric with nilpotent Ricci operator.

Stable Rank of Group C*-algebras of Some Disconnected Lie Groups

  • Sudo, Takahiro
    • Kyungpook Mathematical Journal
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    • v.47 no.2
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    • pp.203-219
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    • 2007
  • We estimate the stable rank and connected stable rank of group $C^*$-algebra of certain disconnected solvable Lie groups such as semi-direct products of connected solvable Lie groups by the integers.

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CLASSIFICATION OF SOLVABLE LIE GROUPS WHOSE NON-TRIVIAL COADJOINT ORBITS ARE OF CODIMENSION 1

  • Ha, Hieu Van;Hoa, Duong Quang;Le, Vu Anh
    • Communications of the Korean Mathematical Society
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    • v.37 no.4
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    • pp.1181-1197
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    • 2022
  • We give a complete classification of simply connected and solvable real Lie groups whose nontrivial coadjoint orbits are of codimension 1. This classification of the Lie groups is one to one corresponding to the classification of their Lie algebras. Such a Lie group belongs to a class, called the class of MD-groups. The Lie algebra of an MD-group is called an MD-algebra. Some interest properties of MD-algebras will be investigated as well.

STABLE RANK OF TWISTED CROSSED PRODUCTS OF $C^{*}-ALGEBRAS$ BY ABELIAN GROUPS

  • Sudo, Takahiro
    • The Pure and Applied Mathematics
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    • v.10 no.2
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    • pp.103-118
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    • 2003
  • We estimate the stable rank of twisted crossed products of $C^{*}-algebras$ by topological Abelian groups. As an application we estimate the stable rank of twisted crossed products of $C^{*}-algebras$ by solvable Lie groups. In particular, we obtain the stable rank estimate of twisted group $C^{*}-algebras$ of solvable Lie groups by the (reduced) dimension and (generalized) rank of groups.

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The Geometry Descriptions of Crystallographic Groups of Sol41

  • Yoo, Won Sok
    • Journal of Integrative Natural Science
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    • v.10 no.2
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    • pp.110-114
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    • 2017
  • The connected and simply connected four-dimensional matrix solvable Lie group $Sol^4_1$ is the four-dimensional geometry. A crystallographic group of $Sol^4_1$ is a discrete cocompact subgroup of $Sol^4_1{\rtimes}D(4)$. In this paper, we geometrically describe the crystallographic groups of $Sol^4_1$.

RIEMANNIAN SUBMERSIONS OF SO0(2, 1)

  • Byun, Taechang
    • Journal of the Korean Mathematical Society
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    • v.58 no.6
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    • pp.1407-1419
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    • 2021
  • The Iwasawa decomposition NAK of the Lie group G = SO0(2, 1) with a left invariant metric produces Riemannian submersions G → N\G, G → A\G, G → K\G, and G → NA\G. For each of these, we calculate the curvature of the base space and the lifting of a simple closed curve to the total space G. Especially in the first case, the base space has a constant curvature 0; the holonomy displacement along a (null-homotopic) simple closed curve in the base space is determined only by the Euclidean area of the region surrounded by the curve.

LEFT-INVARIANT MINIMAL UNIT VECTOR FIELDS ON THE SEMI-DIRECT PRODUCT Rn

  • Yi, Seung-Hun
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.5
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    • pp.951-960
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    • 2010
  • We provide the set of left-invariant minimal unit vector fields on the semi-direct product $\mathbb{R}^n\;{\rtimes}_p\mathbb{R}$, where P is a nonsingular diagonal matrix and on the 7 classes of 4-dimensional solvable Lie groups of the form $\mathbb{R}^3\;{\rtimes}_p\mathbb{R}$ which are unimodular and of type (R).

ON WEAKLY EINSTEIN ALMOST CONTACT MANIFOLDS

  • Chen, Xiaomin
    • Journal of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.707-719
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    • 2020
  • In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a (2n + 1)-dimensional Sasakian manifold admits a weakly Einstein metric, then its scalar curvature s satisfies -6 ⩽ s ⩽ 6 for n = 1 and -2n(2n + 1) ${\frac{4n^2-4n+3}{4n^2-4n-1}}$ ⩽ s ⩽ 2n(2n + 1) for n ⩾ 2. Secondly, for a (2n + 1)-dimensional weakly Einstein contact metric (κ, μ)-manifold with κ < 1, we prove that it is flat or is locally isomorphic to the Lie group SU(2), SL(2), or E(1, 1) for n = 1 and that for n ⩾ 2 there are no weakly Einstein metrics on contact metric (κ, μ)-manifolds with 0 < κ < 1. For κ < 0, we get a classification of weakly Einstein contact metric (κ, μ)-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic (κ, μ)-manifold with κ < 0 is locally isomorphic to a solvable non-nilpotent Lie group.