In this paper we study ad-nilpotent elements in Lie algebras arising from semiprime associative rings R free of 2-torsion. With the idea of keeping under control the torsion of R we introduce a more restrictive notion of ad-nilpotent element, pure ad-nilpotent element, which is a only technical condition since every ad-nilpotent element can be expressed as an orthogonal sum of pure ad-nilpotent elements of decreasing indices. This allows us to be more precise when setting the torsion inside the ring R in order to describe its ad-nilpotent elements. If R is a semiprime ring and a ∈ R is a pure adnilpotent element of R of index n with R free of t and n t -torsion for t = [ n+1 2 ], then n is odd and there exists λ ∈ C(R) such that a − λ is nilpotent of index t. If R is a semiprime ring with involution * and a is a pure ad-nilpotent element of Skew(R, * ) free of t and n t -torsion for t = [ n+1 2 ], then either a is an adnilpotent element of R of the same index n (this may occur if n ≡ 1, 3 (mod 4)) or R is a nilpotent element of R of index t + 1 and R satisfies a nontrivial GPI (this may occur if n ≡ 0, 3 (mod 4)). The case n ≡ 2 (mod 4) is not possible.
Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials.
Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second-order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second-order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials.
In this paper, we study ad-nilpotent elements of semiprime rings R with involution * whose indices of ad-nilpotence differ on Skew(R, * ) and R. The existence of such an ad-nilpotent element a implies the existence of a GPI of R, and determines a big part of its structure. When moving to the symmetric Martindale ring of quotientsThere exists an idempotent e ∈ Q s m (R) that orthogonally decomposes a = ea +(1−e)a and either ea and (1 − e)a are ad-nilpotent of the same index (in this case the index of adnilpotence of a in Skew(Q s m (R), * ) is congruent with 0 modulo 4), or ea and (1 − e)a have different indices of ad-nilpotence (in this case the index of ad-nilpotence of a in Skew(Q s m (R), * ) is congruent with 3 modulo 4). Furthermore, we show that Q s m (R) has a finite Z-grading induced by a * -complete family of orthogonal idempotents and that eQ s m (R)e, which contains ea, is isomorphic to a ring of matrices over its Communicated by Shiping Liu.
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