# Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an overview

@article{Terwilliger2001TwoLT, title={Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an overview}, author={Paul M. Terwilliger}, journal={arXiv: Rings and Algebras}, year={2001} }

## 243 Citations

### Some algebra related to P- and Q-polynomial association schemes

- MathematicsCodes and Association Schemes
- 1999

A mild generalization of a Leonard pair called a tridiagonal pair, such that for each transformation all eigenspaces have dimension one.

### Tridiagonal pairs and the quantum affine algebra $U_q({\hat {sl}}_2)$

- Mathematics
- 2003

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. By definition a Leonard pair on $V$ is a pair of linear transformations $A:V\to V$ and $A^*:V\to V$…

### Two linear transformations each tridiagonal with respect to an eigenbasis of the other: comments on the split decomposition

- Mathematics
- 2003

### Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

- Mathematics
- 2006

### Spin Leonard pairs

- Mathematics
- 2007

Let $${\mathbb K}$$ denote a field, and let V denote a vector space over $${\mathbb K}$$ of finite positive dimension. A pair A, A* of linear operators on V is said to be a Leonard pair on V whenever…

## References

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### Some algebra related to P- and Q-polynomial association schemes

- MathematicsCodes and Association Schemes
- 1999

A mild generalization of a Leonard pair called a tridiagonal pair, such that for each transformation all eigenspaces have dimension one.

### Two linear transformations each tridiagonal with respect to an eigenbasis of the other: comments on the split decomposition

- Mathematics
- 2003

### Leonard Pairs from 24 Points of View

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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)…

### LEONARD PAIRS AND THE ASKEY-WILSON RELATIONS

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Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A*:V→V which satisfy the following two…

### Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations

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- 2001

We define an algebra on two generators which we call the Tridiagonal algebra, and we consider its irreducible modules. The algebra is defined as follows. Let K denote a field, and let $\beta, \gamma,…

### The Terwilliger Algebra of the Hypercube

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An elementary proof that QD has the Q -polynomial property is given and T is a homomorphic image of the universal enveloping algebra of the Lie algebrasl2 (C).