Star-Shapedness and K-Orbits in Complex Semisimple Lie Algebras

AbstractGiven a complex semisimple Lie algebra is a compact real form of g), let be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra , where t is a maximal abelian subalgebra of . Given x ∈ g, we consider π(Ad(K)x), where K is the analytic subgroup G corresponding to , and show that it is star-shaped. The result extends a result of Tsing. We also consider the generalized numerical range f (Ad(K)x), where f is a linear functional on g. We establish the star-shapedness of f (Ad(K)x) for simple Lie algebras of type B.

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Further results on q-Lie groups, q-Lie algebras and q-homogeneous spaces

Special Matrices ◽

10.1515/spma-2020-0129 ◽

2021 ◽

Vol 9 (1) ◽

pp. 119-148

Author(s):

Thomas Ernst

Keyword(s):

Weyl Group ◽

Lie Groups ◽

Lie Algebra ◽

Lie Algebras ◽

Cartan Subalgebra ◽

Homogeneous Spaces ◽

Iwasawa Decomposition ◽

Base Field ◽

Killing Form ◽

Exact Sequences

Abstract We introduce most of the concepts for q-Lie algebras in a way independent of the base field K. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q-roots for certain well-known q-Lie groups contain an extra q-addition, and consequently, for most of the quantities which are q-deformed, we add a prefix q in the respective name. Important examples are the q-Cartan subalgebra and the q-Cartan Killing form. We introduce the concept q-homogeneous spaces in a formal way exemplified by the examples S U q ( 1 , 1 ) S O q ( 2 ) {{S{U_q}\left( {1,1} \right)} \over {S{O_q}\left( 2 \right)}} and S O q ( 3 ) S O q ( 2 ) {{S{O_q}\left( 3 \right)} \over {S{O_q}\left( 2 \right)}} with corresponding q-Lie groups and q-geodesics. By introducing a q-deformed semidirect product, we can define exact sequences of q-Lie groups and some other interesting q-homogeneous spaces. We give an example of the corresponding q-Iwasawa decomposition for SLq(2).

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KRULL DIMENSION OF AFFINOID ENVELOPING ALGEBRAS OF SEMISIMPLE LIE ALGEBRAS

Glasgow Mathematical Journal ◽

10.1017/s0017089513000487 ◽

2013 ◽

Vol 55 (A) ◽

pp. 7-26

Author(s):

KONSTANTIN ARDAKOV ◽

IAN GROJNOWSKI

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Krull Dimension ◽

Semisimple Lie Algebra ◽

Semisimple Lie Algebras ◽

Enveloping Algebras ◽

Enveloping Algebra ◽

Complex Semisimple

AbstractUsing Beilinson–Bernstein localisation, we give another proof of Levasseur's theorem on the Krull dimension of the enveloping algebra of a complex semisimple Lie algebra. The proof also extends to the case of affinoid enveloping algebras.

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QCD breaks Lorentz invariance and colour

Modern Physics Letters A ◽

10.1142/s0217732316500607 ◽

2016 ◽

Vol 31 (10) ◽

pp. 1650060 ◽

Cited By ~ 8

Author(s):

A. P. Balachandran

Keyword(s):

Quantum Chromodynamics ◽

Lorentz Group ◽

Quantum Electrodynamics ◽

Cartan Subalgebra ◽

Lorentz Invariance ◽

Commun Math Phys ◽

Maximal Abelian Subalgebra ◽

Poincare Group ◽

Abelian Subalgebra ◽

Math Phys

In the previous work [A. P. Balachandran and S. Vaidya, Eur. Phys. J. Plus 128, 118 (2013)], we have argued that the algebra of non-Abelian superselection rules is spontaneously broken to its maximal Abelian subalgebra, that is, the algebra generated by its completing commuting set (the two Casimirs, isospin and a basis of its Cartan subalgebra). In this paper, alternative arguments confirming these results are presented. In addition, Lorentz invariance is shown to be broken in quantum chromodynamics (QCD), just as it is in quantum electrodynamics (QED). The experimental consequences of these results include fuzzy mass and spin shells of coloured particles like quarks, and decay life times which depend on the frame of observation [D. Buchholz, Phys. Lett. B 174, 331 (1986); D. Buchholz and K. Fredenhagen, Commun. Math. Phys. 84, 1 (1982; J. Fröhlich, G. Morchio and F. Strocchi, Phys. Lett. B 89, 61 (1979); A. P. Balachandran, S. Kürkçüoğlu, A. R. de Queiroz and S. Vaidya, Eur. Phys. J. C 75, 89 (2015); A. P. Balachandran, S. Kürkçüoğlu and A. R. de Queiroz, Mod. Phys. Lett. A 28, 1350028 (2013)]. In a paper under preparation, these results are extended to the ADM Poincaré group and the local Lorentz group of frames. The renormalisation of the ADM energy by infrared gravitons is also studied and estimated.

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On split involutive regular BiHom-Lie superalgebras

Open Mathematics ◽

10.1515/math-2020-0144 ◽

2020 ◽

Vol 18 (1) ◽

pp. 476-485

Author(s):

Shuangjian Guo ◽

Xiaohui Zhang ◽

Shengxiang Wang

Keyword(s):

Lie Algebras ◽

Lie Superalgebra ◽

Natural Generalization ◽

Lie Superalgebras ◽

Maximal Length ◽

Maximal Abelian Subalgebra ◽

Abelian Subalgebra

Abstract The goal of this paper is to examine the structure of split involutive regular BiHom-Lie superalgebras, which can be viewed as the natural generalization of split involutive regular Hom-Lie algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular BiHom-Lie superalgebra {\mathfrak{L}} is of the form {\mathfrak{L}}=U+{\sum }_{\alpha }{I}_{\alpha } with U a subspace of a maximal abelian subalgebra H and any I α , a well-described ideal of {\mathfrak{L}} , satisfying [I α , I β ] = 0 if [α] ≠ [β]. In the case of {\mathfrak{L}} being of maximal length, the simplicity of {\mathfrak{L}} is also characterized in terms of connections of roots.

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Vanishing resonance and representations of Lie algebras

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2013-0073 ◽

2015 ◽

Vol 2015 (706) ◽

Author(s):

Stefan Papadima ◽

Alexander I. Suciu

Keyword(s):

Representation Theory ◽

Lie Algebra ◽

Lie Algebras ◽

Classical Representation ◽

Semisimple Lie Algebra ◽

Representations Of Lie Algebras ◽

Complex Semisimple ◽

Resonance Varieties

AbstractWe explore a relationship between the classical representation theory of a complex, semisimple Lie algebra 𝔤 and the resonance varieties

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Hereditary automorphic Lie algebras

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500767 ◽

2019 ◽

Vol 22 (08) ◽

pp. 1950076

Author(s):

Vincent Knibbeler ◽

Sara Lombardo ◽

Jan A. Sanders

Keyword(s):

Lie Algebras ◽

Cartan Subalgebra ◽

Dimensional Space ◽

Projective Line ◽

Killing Form ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Novel Approach ◽

Automorphic Functions

We show that automorphic Lie algebras which contain a Cartan subalgebra with a constant-spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianization of the automorphic Lie algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite-dimensional space of said equivariant vectors to a finite-dimensional linear computation and the determination of the ring of automorphic functions on the projective line.

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On a chain of reproducing kernel Cartan subalgebras

Moroccan Journal of Pure and Applied Analysis ◽

10.2478/mjpaa-2021-0005 ◽

2021 ◽

Vol 7 (1) ◽

pp. 43-49

Author(s):

Anoh Yannick Kraidi ◽

Kinvi Kangni

Keyword(s):

Lie Algebra ◽

Cartan Subalgebra ◽

Reproducing Kernel ◽

Killing Form ◽

Semisimple Lie Algebra ◽

Cartan Subalgebras ◽

A Chain

AbstractLet 𝔤 be a semisimple Lie algebra, j a Cartan subalgebra of 𝔤, j*, the dual of j, jv the bidual of j and B(., .) the restriction to j of the Killing form of 𝔤. In this work, we will construct a chain of reproducing kernel Cartan subalgebras ordered by inclusion.

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