English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

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 pairon $V$ is an ordered pair of linear transformations in End$(V)$ such that foreach of these transformations there exists a basis for $V$ with respect towhich the matrix representing that transformation is diagonal and the matrixrepresenting the other transformation is irreducible tridiagonal. Whenever thetridiagonal matrices are bipartite, the Leonard pair is said to be totallybipartite. A mild weakening yields a type of Leonard pair said to be totallyalmost bipartite. A Leonard pair is said to be totally B/AB whenever it istotally bipartite or totally almost bipartite. The notion of a Leonard tripleand the corresponding notion of totally B/AB are similarly defined. There arefamilies of Leonard pairs and Leonard triples said to have Bannai/Ito type. TheLeonard pairs and Leonard triples of interest to us are totally B/AB and ofBannai/Ito type.  Let $\mathcal{A}$ denote the unital associative $\K$-algebra defined bygenerators $x,y,z$ and relations\[xy+yx=2z,\qquad yz+zy=2x,\qquadzx+xz=2y.\]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 isomorphismthe totally B/AB Leonard pairs of Bannai/Ito type. We classify up toisomorphism the totally B/AB Leonard triples of Bannai/Ito type. We classify upto isomorphism the finite-dimensional irreducible $\mathcal{A}$-modules. Weshow that these three classes of objects are essentially in one-to-onecorrespondence, and describe these correspondences in detail.

Totally Bipartite/ABipartite Leonard pairs and Leonard triples of Bannai/Ito type