scholarly journals On the Killing form of Lie Algebras in Symmetric Ribbon Categories

2015 ◽  
Author(s):  
Igor Buchberger
Keyword(s):  
Lie Algebras ◽  
Killing Form

On Derivations of Lie Algebras

1976 ◽  
Vol 28 (1) ◽  
pp. 174-180 ◽  
Author(s):  
Stephen Berman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Characteristic Zero ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Cartan Matrices

A well known result in the theory of Lie algebras, due to H. Zassenhaus, states that if is a finite dimensional Lie algebra over the field K such that the killing form of is non-degenerate, then the derivations of are all inner, [3, p. 74]. In particular, this applies to the finite dimensional split simple Lie algebras over fields of characteristic zero. In this paper we extend this result to a class of Lie algebras which generalize the split simple Lie algebras, and which are defined by Cartan matrices (for a definition see § 1). Because of the fact that the algebras we consider are usually infinite dimensional, the method we employ in our investigation is quite different from the standard one used in the finite dimensional case, and makes no reference to any associative bilinear form on the algebras.

KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank

1982 ◽  
Vol 34 (6) ◽  
pp. 1215-1239 ◽  
Author(s):  
L. J. Santharoubane
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Maximal Rank ◽  
Killing Form ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Infinite Dimensional ◽  
Dimensional Version

Introduction. The natural problem of determining all the Lie algebras of finite dimension was broken in two parts by Levi's theorem:1) the classification of semi-simple Lie algebras (achieved by Killing and Cartan around 1890)2) the classification of solvable Lie algebras (reduced to the classification of nilpotent Lie algebras by Malcev in 1945 (see [10])).The Killing form is identically equal to zero for a nilpotent Lie algebra but it is non-degenerate for a semi-simple Lie algebra. Therefore there was a huge gap between those two extreme cases. But this gap is only illusory because, as we will prove in this work, a large class of nilpotent Lie algebras is closely related to the Kac-Moody Lie algebras. These last algebras could be viewed as infinite dimensional version of the semisimple Lie algebras.

MATHEMATICA™ PACKAGES FOR COMPUTING PRINCIPAL DECOMPOSITIONS OF SIMPLE LIE ALGEBRAS AND APPLICATIONS IN EXTENDED CONFORMAL FIELD THEORIES

2003 ◽  
Vol 14 (01) ◽  
pp. 1-27 ◽  
Author(s):  
DANIELA GĂRĂJEU ◽  
MIHAIL GĂRĂJEU
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Adjoint Representation ◽  
Cartan Matrix ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Conformal Field

In this article, we propose two Mathematica™ packages for doing calculations in the domain of classical simple Lie algebras. The main goal of the first package, [Formula: see text], is to determine the principal three-dimensional subalgebra of a simple Lie algebra. The package provides several functions which give some elements related to simple Lie algebras (generators in fundamental and adjoint representation, roots, Killing form, Cartan matrix, etc.). The second package, [Formula: see text], concerns the principal decomposition of a Lie algebra with respect to the principal three-dimensional embedding. These packages have important applications in extended two-dimensional conformal field theories. As an example, we present an application in the context of the theory of W-gravity.

scholarly journals On Killing Forms and Invariant Forms of Lie-Yamaguti Superalgebras

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Patricia L. Zoungrana ◽  
A. Nourou Issa
Keyword(s):  
Lie Algebras ◽  
Invariant Form ◽  
Killing Form

The notions of the Killing form and invariant form in Lie algebras are extended to the ones in Lie-Yamaguti superalgebras and some of their properties are investigated. These notions are alsoZ2-graded generalizations of the ones in Lie-Yamaguti algebras.

scholarly journals Invariant quadratic forms on finite dimensional lie algebras

1986 ◽  
Vol 33 (1) ◽  
pp. 21-36 ◽  
Author(s):  
Karl H. Hofmann ◽  
Verena S. Keith
Keyword(s):  
Lie Algebras ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
General Structure ◽  
Killing Form ◽  
Trace Forms ◽  
Finite Dimensional ◽  
Recent Developments ◽  
Finite Dimensional Lie Algebras ◽  
The Ideal

Trace forms have been well studied as invariant quadratic forms on finite dimensional Lie algebras; the best known of these forms in the Cartan-Killing form. All those forms, however, have the ideal [L, L] ∩ R (with the radical R) in the orthogonal L⊥ and thus are frequently degenerate. In this note we discuss a general construction of Lie algebras equipped with non-degenerate quadratic forms which cannot be obtained by trace forms, and we propose a general structure theorem for Lie algebras supporting a non-degenerate invariant quadratic form. These results complement and extend recent developments of the theory of invariant quadratic forms on Lie algebras by Hilgert and Hofmann [2], keith [4], and Medina and Revoy [7].

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).

Mutually unbiased bases and orthogonal decompositions of Lie algebras

2007 ◽  
Vol 7 (4) ◽  
pp. 371-382
Author(s):  
P.O. Boykin ◽  
M. Sitharam ◽  
P.H. Tiep ◽  
P. Wocjan
Keyword(s):  
Lie Algebras ◽  
Open Problem ◽  
Orthogonal Decomposition ◽  
Mutually Unbiased Bases ◽  
Killing Form ◽  
Unitary Matrices ◽  
Complete Collection ◽  
Cartan Subalgebras ◽  
Orthogonal Decompositions ◽  
Adjoint Operation

We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of $\mu$ MUBs in $\K^n$ gives rise to a collection of $\mu$ Cartan subalgebras of the special linear Lie algebra $sl_n(\K)$ that are pairwise orthogonal with respect to the Killing form, where $\K=\R$ or $\K=\C$. In particular, a complete collection of MUBs in $\C^n$ gives rise to a so-called orthogonal decomposition (OD) of $sl_n(\C)$. The converse holds if the Cartan subalgebras in the OD are also $\dag$-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result of \cite{bbrv02} relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. This connection implies that for $n\le 5$ an essentially unique complete collection of MUBs exists. We define \emph{monomial MUBs}, a class of which all known MUB constructions are members, and use the above connection to show that for $n=6$ there are at most three monomial MUBs.

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.

Sign in / Sign up

Export Citation Format

Share Document