scholarly journals On a chain of reproducing kernel Cartan subalgebras

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.

scholarly journals Cartan Subalgebras of Jordan Algebras

1966 ◽  
Vol 27 (2) ◽  
pp. 591-609 ◽  
Author(s):  
N. Jacobson
Keyword(s):  
Lie Algebra ◽  
Jordan Algebra ◽  
Cartan Subalgebra ◽  
Jordan Algebras ◽  
Classical Notion ◽  
Cartan Subalgebras ◽  
Definition Of

In this paper we shall give a definition of an analogue for Jordan algebras of the classical notion of a Cartan subalgebra of a Lie algebra. This is based on a notion of associator nilpotency of a Jordan algebra. A Jordan algebra is called associator nilpotent if there exists a positive (odd) integer M such that every associator of order M formed of elements of is 0 (§2).

Invariant Polynomials of Weyl Groups and Applications to the Centres of Universal Enveloping Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 583-592 ◽  
Author(s):  
C. Y. Lee
Keyword(s):  
Lie Algebra ◽  
Weyl Groups ◽  
Killing Form ◽  
Dimensional Representation ◽  
Universal Enveloping Algebras ◽  
Semisimple Lie Algebra ◽  
Finite Dimensional Representation ◽  
Polynomial Invariants ◽  
Finite Dimensional ◽  
Complete Set

An element in the centre of the universal enveloping algebra of a semisimple Lie algebra was first constructed by Casimir by means of the Killing form. By Schur's lemma, in an irreducible finite-dimensional representation elements in the centre are represented by diagonal matrices of all whose eigenvalues are equal. In section 2 of this paper, we show the existence of a complete set of generators whose eigenvalues in an irreducible representation are closely related to polynomial invariants of the Weyl group W of the Lie algebra (Theorem 1).

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

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).

ISOTROPY REPRESENTATIONS, EIGENVALUES OF A CASIMIR ELEMENT, AND COMMUTATIVE LIE SUBALGEBRAS

2001 ◽  
Vol 64 (1) ◽  
pp. 61-80 ◽  
Author(s):  
DMITRI I. PANYUSHEV
Keyword(s):  
Lie Algebra ◽  
Finite Order ◽  
Killing Form ◽  
Maximal Eigenvalue ◽  
Isotropy Representation ◽  
Semisimple Lie Algebra ◽  
Casimir Element ◽  
Root Of Unity ◽  
Exterior Powers

Let [hfr ] be a reductive subalgebra of a semisimple Lie algebra [gfr ] and C[hfr ] ∈ U([hfr ]) be the Casimir element determined by the restriction of the Killing form on [gfr ] to [hfr ]. The paper studies eigenvalues of C[hfr ] on the isotropy representation [mfr ]≃[gfr ]/[hfr ]. Some general estimates connecting the eigenvalues and the Dynkin indices of [mfr ] are given. If [hfr ] is a symmetric subalgebra, it is shown that describing the maximal eigenvalue of C[hfr ] on exterior powers of [mfr ] is connected with possible dimensions of commutative Lie subalgebras in [mfr ], thereby extending a result of Kostant. In this situation, a formula is also given for the maximal eigenvalue of C[hfr ] on ∧ [mfr ]. More generally, a similar picture arises if [hfr ] = [gfr ]Θ, where Θ is an automorphism of finite order m and [mfr ] is replaced by the eigenspace of Θ corresponding to a primitive mth root of unity.

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

2011 ◽  
Vol 54 (1) ◽  
pp. 44-55
Author(s):  
Wai-Shun Cheung ◽  
Tin-Yau Tam
Keyword(s):  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Numerical Range ◽  
Killing Form ◽  
Semisimple Lie Algebra ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Complex Semisimple ◽  
Analytic Subgroup ◽  
Generalized Numerical Range

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.

On Maximal Subsystems of Root Systems

1968 ◽  
Vol 20 ◽  
pp. 555-574 ◽  
Author(s):  
Nolan R. Wallach
Keyword(s):  
Root System ◽  
Lie Algebra ◽  
Cartan Subalgebra ◽  
Root Systems ◽  
Closed Field ◽  
Semisimple Lie Algebra ◽  
Algebraically Closed Field

Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic 0. Let h be a Cartan subalgebra of g and let Δ be the root system of g with respect to h.

On the tensor system of a semisimple Lie algebra

1972 ◽  
Vol 71 (2) ◽  
pp. 211-226 ◽  
Author(s):  
Timothy Murphy
Keyword(s):  
Lie Algebra ◽  
Point Of View ◽  
Killing Form ◽  
Essential Characteristic ◽  
Semisimple Lie Algebra ◽  
The Subject

AbstractFrom the tensorial point of view, the essential characteristic of a semisimple lie algebra A is the non-singularity of the Killing form, since this enables one to construct a system of Cartesian tensors over A. That system is the subject of this paper.

ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

1992 ◽  
Vol 07 (23) ◽  
pp. 2077-2085 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Scalar Field ◽  
Euclidean Space ◽  
Lie Algebra ◽  
Linear Equations ◽  
Dual Solutions ◽  
Gauge Fields ◽  
System Of Equations ◽  
Semisimple Lie Algebra ◽  
Yang Mills ◽  
Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

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