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M. Ceballos Departamento de Geometrґ y Topologґ Facultad de Matemґ ia ia, aticas, Universidad de Sevilla Sevilla, Spain mceballosgonzalez@gmail.com ґ~ J. Nunez Departamento de Geometrґ y Topologґ Facultad de Matemґ ia ia, aticas, Universidad de Sevilla Sevilla, Spain jnvaldes@us.es ґ A. F. Tenorio Departamento de Economґ Mґ dos Cuantitativos e Ha Econґ ia, eto omica, Escuela Politґ ecnica Superior, Universidad Pablo de Olavide Sevilla, Spain aftenorio@upo.es

Computing maximal Ab elian dimensions in upp er-triangular matrix algebras
The topic which is dealt in this paper is the maximal abelian dimension of a given finitedimensional Lie algebra g, that is, the maximum among the dimensions of the abelian Lie subalgebras of g. More concretely, this maximum is computed for the Lie algebra hn , formed by all the n в n upper-triangular matrices. In this way, every vector in hn can be expressed as follows: x11 x12 · · · x1n 0 x22 · · · x2n hn (xr,s ) = . . .. .. . . . . . . 0 ··· 0 xnn Starting from this expression for the vectors in hn , the following basis can be obtained: B = {Xi,j = hn (xr,s ) | 1 i j n}, where: x
r,s

=

1, 0,

if (r, s) = (i, j ), if (r, s) = (i, j ).

Therefore, the dimension of this algebra is: n(n + 1) , 2 and the nonzero brackets with respect to the basis B are: dim(hn ) = dhn = [Xi,j , Xj,k ] = Xi,k , i = 1 . . . n - 2, j = i + 1 . . . n - 1, k = j + 1 . . . n. [Xi,i , Xi,j ] = Xi,j , j > i. [Xk,i , Xi,i ] = Xk,i , k < i. Apart from other reasons related to Physics, our interest for studying the Lie algebras hn lies in the fact which every finite-dimensional solvable Lie algebra is isomorphic to a Lie subalgebra in some Lie algebra hn [1, Proposition 3.7.3]. Therefore, the computation of the maximal abelian dimension for hn can be considered a first step to study the maximal abelian dimension of any given finite-dimensional solvable Lie algebra. The present paper continues the authors' previous paper [2] in which some properties of the maximal abelian dimension were studied for the algebra hn and a value for its maximal abelian dimension was conjectured: 1


Conjecture. Fixed and given n N \ {1}, the maximal abelian dimension of hn is: if n < 4, n, k 2 + 1, if n = 2k , n 4, M(hn ) = 2 k + k + 1, if n = 2k + 1, n 4. This conjecture was achieved starting from an algorithmic procedure to compute abelian subalgebras in the Lie algebra hn . In fact, this conjecture was already proved for the particular cases n = 2 and n = 3 in [2]. In this paper, we show a proof for the previously commented conjecture for all n N \ {1}. To do it, two lemmas will be proved and applied in order to prove the veracity of the conjecture. To get the proof, the vectors in a given basis of hn have to be distinguished between main vectors and non-main ones for a given basis of the subalgebra. Such a distinction is based on writing each vector in the basis of the subalgebra as a linear combination of the basis of hn ; then these coefficients are written as the rows in a matrix and the vectors corresponding to the pivot positions of its echalon form are the main vectors. References [1] V.S. Varadara jan, Lie Groups, Lie Algebras and Their Representations, Springer, New York, 1984. [2] M. Ceballos, J. Nunez, A.F. Tenorio, Obstructions to represent abelian Lie subalgebras ґ~ in the Lie algebra of upper-triangular matrices. First French-Spanish Congress of Mathematics, Saragossa (Spain), July 2007.

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