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Totally Bipartite/ABipartite Leonard pairs and Leonard triples of Bannai/Ito type | | George M. F. Brown
; | | Date: |
20 Dec 2011 | | Abstract: | This paper is about three classes of objects: Leonard pairs, Leonard triples,
and the finite-dimensional irreducible modules for an algebra $mathcal{A}$.
Let $K$ denote an algebraically closed field of characteristic zero. Let $V$
denote a vector space over $K$ with finite positive dimension. A Leonard pair
on $V$ is an ordered pair of linear transformations in End$(V)$ such that for
each of these transformations there exists a basis for $V$ with respect to
which the matrix representing that transformation is diagonal and the matrix
representing the other transformation is irreducible tridiagonal. Whenever the
tridiagonal matrices are bipartite, the Leonard pair is said to be totally
bipartite. A mild weakening yields a type of Leonard pair said to be totally
almost bipartite. A Leonard pair is said to be totally B/AB whenever it is
totally bipartite or totally almost bipartite. The notion of a Leonard triple
and the corresponding notion of totally B/AB are similarly defined. There are
families of Leonard pairs and Leonard triples said to have Bannai/Ito type. The
Leonard pairs and Leonard triples of interest to us are totally B/AB and of
Bannai/Ito type.
Let $mathcal{A}$ denote the unital associative $K$-algebra defined by
generators $x,y,z$ and relations[xy+yx=2z,qquad yz+zy=2x,qquad
zx+xz=2y.]The algebra $mathcal{A}$ has a presentation involving generators
$x,y$ and relations[x^{2}y+2xyx+yx^{2}=4y,qquad y^{2}x+2yxy+xy^{2}=4x.]
In this paper we obtain the following results. We classify up to isomorphism
the totally B/AB Leonard pairs of Bannai/Ito type. We classify up to
isomorphism the totally B/AB Leonard triples of Bannai/Ito type. We classify up
to isomorphism the finite-dimensional irreducible $mathcal{A}$-modules. We
show that these three classes of objects are essentially in one-to-one
correspondence, and describe these correspondences in detail. | | Source: | arXiv, 1112.4577 | | Services: | Forum | Review | PDF | Favorites | |
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