منابع مشابه
2 8 Ju n 20 04 Leonard pairs from 24 points of view ∗
Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V that satisfy both conditions below: (i) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A is irreducible tridiagonal. (ii) There exists a basis for V with respect to wh...
متن کاملBalanced Leonard Pairs
Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A∗ : V → V that satisfy the following two conditions: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal. (ii) There ex...
متن کامل“Leonard Pairs” in Classical Mechanics
Ġ = {G,H}. In particular, the DV F is called an integral if it has zero PB with the Hamiltonian {F,H} = 0. In this case F does not depend on t. In many problems of the classical mechanics DV form elegant algebraic structures which are closed with respect to PB. The Poisson structures with non-linear PB were discussed in [9] and [6]. Sklyanin introduced [9] the so-called quadratic Poisson algebr...
متن کاملNormalized Leonard pairs and Askey-Wilson relations
Let V denote a vector space with finite positive dimension, and let (A,A∗) denote a Leonard pair on V . As is known, the linear transformations A, A∗ satisfy the Askey-Wilson relations A 2 A ∗ − βAA ∗ A+A∗A2 − γ (AA∗+A∗A)− ̺A∗ = γ∗A2 + ωA+ η I, A ∗2 A− βA ∗ AA ∗+ AA∗2− γ∗(A∗A+AA∗)− ̺∗A = γA∗2+ ωA∗+ η∗I, for some scalars β, γ, γ∗, ̺, ̺∗, ω, η, η∗. The scalar sequence is unique if the dimension of V ...
متن کاملAskey-Wilson relations and Leonard pairs
It is known that if (A,A∗) is a Leonard pair, then the linear transformations A, A∗ satisfy the Askey-Wilson relations A 2 A ∗ − βAA ∗ A + A∗A2 − γ (AA∗+A∗A) − ̺A∗ = γ∗A2 + ωA + η I, A ∗2 A− βA ∗ AA ∗+ AA∗2− γ∗(A∗A+AA∗) − ̺∗A = γA∗2+ ωA∗+ η∗I, for some scalars β, γ, γ∗, ̺, ̺∗, ω, η, η∗. The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pai...
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ژورنال
عنوان ژورنال: Rocky Mountain Journal of Mathematics
سال: 2002
ISSN: 0035-7596
DOI: 10.1216/rmjm/1030539699