scholarly journals The spectral radius of the Coxeter transformations for a generalized Cartan matrix

1994 ◽  
Vol 300 (1) ◽  
pp. 331-339 ◽  
Author(s):  
Claus Michael Ringel
Keyword(s):  
Spectral Radius ◽  
Cartan Matrix ◽  
Generalized Cartan Matrix

Euclidean Lie Algebras

1969 ◽  
Vol 21 ◽  
pp. 1432-1454 ◽  
Author(s):  
Robert V. Moody
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Cartan Matrix ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Generalized Cartan Matrix

Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras. We were able to show that E((A ij)) is simple providing that (A ij) does not fall in the list given in (4, Table). We did not prove the converse, however.The diagrams of the table of (4) appear in Table 2. Call the matrices that they represent Euclidean matrices and their corresponding algebras Euclidean Lie algebras. Our first objective is to show that Euclidean Lie algebras are not simple.

Symmetrizable Intersection Matrix Lie Algebras

2011 ◽  
Vol 18 (04) ◽  
pp. 639-646 ◽  
Author(s):  
Mang Xu ◽  
Liangang Peng
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Matrix ◽  
Intersection Matrix ◽  
Special Cases ◽  
Generalized Cartan Matrix

Associated to every symmetrizable generalized intersection matrix A, we define a Lie algebra, called an SIM-Lie algebra. We prove that SIM-Lie algebras keep unchange under braid-equivalences. Two special cases are considered. In the case when A is a symmetrizable generalized Cartan matrix, we show that the corresponding SIM-Lie algebra is just the Kac–Moody Lie algebra. In another case when A is an intersection matrix, we prove that the corresponding SIM-Lie algebra is just the intersection matrix Lie algebra in the sense of Slodowy.

scholarly journals Monoid gradings on algebras and the cartan determinant conjecture

1998 ◽  
Vol 41 (3) ◽  
pp. 539-551 ◽  
Author(s):  
Manuel Saorín
Keyword(s):  
Artinian Ring ◽  
Positive Answer ◽  
Cartan Matrix ◽  
Commutative Monoid ◽  
Graded Algebras ◽  
Finite Dimensional ◽  
Definite Answer ◽  
Generalized Cartan Matrix ◽  
Finite Dimensional Algebras

In this work we tackle the Cartan determinant conjecture for finite-dimensional algebras through monoid gradings. Given an adequate ∑-grading on the left Artinian ring A, where ∑ is a monoid, we construct a generalized Cartan matrix with entries in ℤ∑, which is right invertitale whenever gl.dim A < ∞. That gives a positive answer to the conjecture when A admits a strongly adequate grading by an aperiodic commutative monoid. We then show that, even though this does not give a definite answer to the conjecture, it strictly widens the class of known graded algebras for which it is true.

On Some Twisted Chevalley Groups Over Laurent Polynomial Rings

1981 ◽  
Vol 33 (5) ◽  
pp. 1182-1201 ◽  
Author(s):  
Jun Morita
Keyword(s):  
Lie Algebra ◽  
Rational Numbers ◽  
Laurent Polynomial ◽  
Cartan Matrix ◽  
Polynomial Rings ◽  
Complex Numbers ◽  
Real Numbers ◽  
Defining Relations ◽  
Image Position ◽  
Generalized Cartan Matrix

We let Z denote the ring of rational integers, Q the field of rational numbers, R the field of real numbers, and C the field of complex numbers.For elements e and f of a Lie algebra, [e,f] denotes the bracket of e and f. A generalized Cartan matrix C = (cij) is a square matrix of integers satisfying cii = 2, cij ≦ 0 if i ≠ j, cij = 0 if and only if cji = 0. For any generalized Cartan matrix C = (cij) of size l × l and for any field F of characteristic zero, denotes the Lie algebra over F generated by 3l generators e1, …, el, h1, …, hl, f1, …, fl with the defining relationsfor all i, j,for distinct i, j.

SPECTRAL RADIUS OF NONSINGULAR TRANSFORMATIONS

1989 ◽  
Vol 15 (1) ◽  
pp. 275
Author(s):  
NADKARNI ◽  
ROBERTSON
Keyword(s):  
Spectral Radius

scholarly journals On Gibbs Measures and Spectra of Ruelle Transfer Operators

2017 ◽  
Vol 60 (2) ◽  
pp. 411-421
Author(s):  
Luchezar Stoyanov
Keyword(s):  
Spectral Radius ◽  
Spectral Properties ◽  
Gibbs Measures ◽  
Transfer Operator ◽  
Frobenius Theorem ◽  
Transfer Operators

AbstractWe prove a comprehensive version of the Ruelle–Perron–Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previously known estimates.

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