Symmetrizable Intersection Matrix Lie Algebras

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.

Monoid gradings on algebras and the cartan determinant conjecture

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

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.

Euclidean Lie Algebras

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.