Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

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Isomorphism of Some Simple 2-Graded Lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1977-031-4 ◽

1977 ◽

Vol 29 (2) ◽

pp. 289-294

Author(s):

Dragomir Ž. Djoković

Keyword(s):

Lie Algebras ◽

Characteristic Zero ◽

Graded Lie Algebras ◽

Finite Dimensional ◽

Definition Of

The grading is by integers modulo 2 and we refer to it as 2-grading. For the definition of 2-graded Lie algebras L and their properties we refer the reader to the papers [1; 2; 3]. All algebras considered here are finite-dimensional over a field F of characteristic zero.

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Euclidean Lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-158-2 ◽

1969 ◽

Vol 21 ◽

pp. 1432-1454 ◽

Cited By ~ 55

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.

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Almost nilpotent Lie algebras

Glasgow Mathematical Journal ◽

10.1017/s0017089500006625 ◽

1987 ◽

Vol 29 (1) ◽

pp. 7-11 ◽

Cited By ~ 3

Author(s):

David A. Towers

Keyword(s):

Lie Algebras ◽

Characteristic Zero ◽

Nilpotent Lie Algebras ◽

Semisimple Lie Algebras ◽

Finite Dimensional ◽

Finite Dimensional Lie Algebras

Throughout we shall consider only finite-dimensional Lie algebras over a field of characteristic zero. In [3] it was shown that the classes of solvable and of supersolvable Lie algebras of dimension greater than two are characterised by the structure of their subalgebra lattices. The same is true of the classes of simple and of semisimple Lie algebras of dimension greater than three. However, it is not true of the class of nilpotent Lie algebras. We seek here the smallest class containing all nilpotent Lie algebras which is so characterised.

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