Upper bound for the dimension of the center of a solvable Lie algebra

2000 ◽  
Vol 43 (2-3) ◽  
pp. 101-107
Author(s):  
J.C. Ndogmo
Keyword(s):  
Lie Algebra ◽  
Upper Bound ◽  
Solvable Lie Algebra

scholarly journals RESIDUAL SYMMETRIES IN THE PRESENCE OF AN EM BACKGROUND

2003 ◽  
Vol 18 (09) ◽  
pp. 629-641 ◽  
Author(s):  
H. L. CARRION ◽  
M. ROJAS ◽  
F. TOPPAN
Keyword(s):  
Lie Algebra ◽  
Symmetry Algebra ◽  
Algebraic Model ◽  
Two Dimensional ◽  
Square Roots ◽  
Residual Symmetry ◽  
Solvable Lie Algebra ◽  
Poincaré Algebra ◽  
Model Independent ◽  
Symmetry Algebras

The symmetry algebra of a QFT in the presence of an external EM background (named "residual symmetry") is investigated within a Lie-algebraic, model-independent scheme. Some results previously encountered in the literature are extended here. In particular we compute the symmetry algebra for a constant EM background in D = 3 and D = 4 dimensions. In D = 3 dimensions the residual symmetry algebra, for generic values of the constant EM background, is isomorphic to [Formula: see text], with [Formula: see text] the centrally extended two-dimensional Poincaré algebra. In D = 4 dimension the generic residual symmetry algebra is given by a seven-dimensional solvable Lie algebra which is explicitly computed. Residual symmetry algebras are also computed for specific non-constant EM backgrounds and in the supersymmetric case for a constant EM background. The supersymmetry generators are given by the "square roots" of the deformed translations.

scholarly journals Simply transitive NIL-affine actions of solvable Lie groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jonas Deré ◽  
Marcos Origlia
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Main Tool ◽  
Full Description ◽  
Nilpotent Lie Group ◽  
Affine Transformations ◽  
Solvable Lie Algebra ◽  
Solvable Lie Group

Abstract Every simply connected and connected solvable Lie group 𝐺 admits a simply transitive action on a nilpotent Lie group 𝐻 via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups 𝐺 can act simply transitively on which Lie groups 𝐻. So far, the focus was mainly on the case where 𝐺 is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action ρ : G → Aff ⁡ ( H ) \rho\colon G\to\operatorname{Aff}(H) is simply transitive by looking only at the induced morphism φ : g → aff ⁡ ( h ) \varphi\colon\mathfrak{g}\to\operatorname{aff}(\mathfrak{h}) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group 𝐺 acts simply transitively on a given nilpotent Lie group 𝐻, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.

scholarly journals A Particular Test Element of a Free Solvable Lie Algebra of Rank Two

2007 ◽  
Vol 37 (4) ◽  
pp. 1315-1326 ◽  
Author(s):  
Ahmet Temizyürek ◽  
Naime Ekici
Keyword(s):  
Lie Algebra ◽  
Test Element ◽  
Solvable Lie Algebra

scholarly journals NILPOTENT SUBSPACES AND NILPOTENT ORBITS

2018 ◽  
Vol 106 (1) ◽  
pp. 104-126
Author(s):  
DMITRI I. PANYUSHEV ◽  
OKSANA S. YAKIMOVA
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Upper Bound ◽  
Nilpotent Orbit ◽  
Orbit Closure ◽  
Maximal Dimension ◽  
Nilpotent Orbits ◽  
Minimal Dimension ◽  
Semisimple Complex ◽  
Stable Subspace

Let $G$ be a semisimple complex algebraic group with Lie algebra $\mathfrak{g}$. For a nilpotent $G$-orbit ${\mathcal{O}}\subset \mathfrak{g}$, let $d_{{\mathcal{O}}}$ denote the maximal dimension of a subspace $V\subset \mathfrak{g}$ that is contained in the closure of ${\mathcal{O}}$. In this note, we prove that $d_{{\mathcal{O}}}\leq {\textstyle \frac{1}{2}}\dim {\mathcal{O}}$ and this upper bound is attained if and only if ${\mathcal{O}}$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V={\textstyle \frac{1}{2}}\dim {\mathcal{O}}$, then $V$ is the nilradical of a polarisation of ${\mathcal{O}}$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_{2}$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits ${\mathcal{O}}$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$, or (2) is the only $B$-stable subspace $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$.

scholarly journals GROUND RING OF α-SYMMETRIES AND SEQUENCE OF RNS STRING THEORIES

2009 ◽  
Vol 24 (31) ◽  
pp. 5897-5924 ◽  
Author(s):  
DIMITRI POLYAKOV
Keyword(s):  
Lie Algebra ◽  
Extra Dimensions ◽  
Space Time ◽  
Field Theories ◽  
Superstring Theory ◽  
Local Gauge ◽  
Work In Progress ◽  
Gauge Symmetries ◽  
Solvable Lie Algebra ◽  
Virasoro Type

We construct a sequence of nilpotent BRST charges in RNS superstring theory, based on new gauge symmetries on the worldsheet, found in this paper. These new local gauge symmetries originate from the global nonlinear space–time α-symmetries, shown to form a noncommutative ground ring in this work. The important subalgebra of these symmetries is U (3) × X6, where X6 is solvable Lie algebra consisting of six elements with commutators reminiscent of the Virasoro type. We argue that the new BRST charges found in this work describe the kinetic terms in string field theories around curved backgrounds of the AdS × CP n-type, determined by the geometries of hidden extra dimensions induced by the global α-generators. The identification of these backgrounds is however left for the work in progress.

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