Lie algebras of an affinization of a generalized Cartan matrix

1998 ◽  
Vol 70 (6) ◽  
pp. 438-446
Author(s):  
Kai Hing Lum
Keyword(s):  
Lie Algebras ◽  
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.

Simple Quotients of Euclidean Lie Algebras

1970 ◽  
Vol 22 (4) ◽  
pp. 839-846 ◽  
Author(s):  
Robert V. Moody
Keyword(s):  
Weyl Group ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Characteristic Zero ◽  
Chain Condition ◽  
Cartan Matrix ◽  
Finite Codimension ◽  
Infinite Dimensional ◽  
Object Of Study ◽  
Ascending Chain Condition

In [2], we considered a class of Lie algebras generalizing the classical simple Lie algebras. Using a field Φ of characteristic zero and a square matrix (Aij) of integers with the properties (1) Aii = 2, (2) Aij ≦ 0 if i ≠ j, (3) Aij = 0 if and only if Ajt = 0, and (4) is symmetric for some appropriate non-zero rational a Lie algebra E = E((Aij)) over Φ can be constructed, together with the usual accoutrements: a root system, invariant bilinear form, and Weyl group.For indecomposable (A ij), E is simple except when (Aij) is singular and removal of any row and corresponding column of (Aij) leaves a Cartan matrix. The non-simple Es, Euclidean Lie algebras, were our object of study in [3] as well as in the present paper. They are infinite-dimensional, have ascending chain condition on ideals, and proper ideals are of finite codimension.

scholarly journals Duality Identities for Moduli Functions of Generalized Melvin Solutions Related to Classical Lie Algebras of Rank 4

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
S. V. Bolokhov ◽  
V. D. Ivashchuk
Keyword(s):  
Lie Algebras ◽  
Toda Chain ◽  
Scalar Fields ◽  
Cartan Matrix ◽  
Radial Coordinate ◽  
Cylindrically Symmetric ◽  
Symmetry Properties ◽  
The Matrix ◽  
Chain Type ◽  
Internal Symmetries

We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank 4 (namely, A4, B4, C4, and D4) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically symmetric gravitational configuration in D dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions Hs(z) (s=1,…,4) of squared radial coordinate z=ρ2 obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers (n1,n2,n3,n4)=(4,6,6,4),(8,14,18,10),(7,12,15,16),(6,10,6,6) for Lie algebras A4, B4, C4, and D4, respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued 4×4 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A4 case) the matrix representing a generator of the Z2-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances. We also calculate 2-form flux integrals over 2-dimensional discs and corresponding Wilson loop factors over their boundaries.

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 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.

scholarly journals On generalized Melvin solutions for Lie algebras of rank 3

2018 ◽  
Vol 15 (07) ◽  
pp. 1850108 ◽  
Author(s):  
S. V. Bolokhov ◽  
V. D. Ivashchuk
Keyword(s):  
Lie Algebras ◽  
Power Law ◽  
Wilson Loop ◽  
Dynkin Diagram ◽  
Scalar Fields ◽  
Cartan Matrix ◽  
Two Dimensional ◽  
Identity Matrix ◽  
Matrix Formula ◽  
Radial Variable

Generalized Melvin solutions for rank-[Formula: see text] Lie algebras [Formula: see text], [Formula: see text] and [Formula: see text] are considered. Any solution contains metric, three Abelian 2-forms and three scalar fields. It is governed by three moduli functions [Formula: see text] ([Formula: see text] and [Formula: see text] is a radial variable), obeying three differential equations with certain boundary conditions imposed. These functions are polynomials with powers [Formula: see text] for Lie algebras [Formula: see text], [Formula: see text], [Formula: see text], respectively. The solutions depend upon integration constants [Formula: see text]. The power-law asymptotic relations for polynomials at large [Formula: see text] are governed by integer-valued [Formula: see text] matrix [Formula: see text], which coincides with twice the inverse Cartan matrix [Formula: see text] for Lie algebras [Formula: see text] and [Formula: see text], while in the [Formula: see text]-case [Formula: see text], where [Formula: see text] is the identity matrix and [Formula: see text] is a permutation matrix, corresponding to a generator of the [Formula: see text]-group of symmetry of the Dynkin diagram. The duality identities for polynomials and asymptotic relations for solutions at large distances are obtained. Two-form flux integrals over a two-dimensional disc of radius [Formula: see text] and corresponding Wilson loop factors over a circle of radius [Formula: see text] are presented.

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.

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