Finite-dimensional irreducible modules of the Bannai–Ito 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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HOPF ALGEBRAS OF DIMENSION 30

Journal of Algebra and Its Applications ◽

10.1142/s0219498810003902 ◽

2010 ◽

Vol 09 (01) ◽

pp. 11-15 ◽

Cited By ~ 2

Author(s):

DAIJIRO FUKUDA

Keyword(s):

Hopf Algebra ◽

Group Algebra ◽

Hopf Algebras ◽

Characteristic Zero ◽

Closed Field ◽

Algebraically Closed Field ◽

Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

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On basic Cycles of An, Bn, Cn and Dn

Canadian Journal of Mathematics ◽

10.4153/cjm-1985-010-8 ◽

1985 ◽

Vol 37 (1) ◽

pp. 122-140 ◽

Cited By ~ 2

Author(s):

D. J. Britten ◽

F. W. Lemire

Keyword(s):

Lie Algebra ◽

Dual Space ◽

Cartan Subalgebra ◽

Closed Field ◽

Generating Set ◽

Enveloping Algebra ◽

Simple Lie Algebra ◽

Finite Dimensional ◽

Irreducible Modules ◽

Positive Roots

In this paper, we investigate a conjecture of Dixmier [2] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L. The basic cycles form a generating set for the cycle subalgebra.First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F, H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L, C(L) the centralizer of H in U(L), Φ the set of nonzero roots in H*, the dual space of H, Δ = {α1, …, αn} a base of Φ, and Φ+ = {β1, …, βm} the positive roots corresponding to Δ.

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