The classification of Leonard triples of Racah type

Let K$\mathbb{K}$ denote an algebraically closed field of characteristic zero. Let V   denote a vector space over K$\mathbb{K}$ with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A  , A^?$A^{?}$, A^ε$A^{\epsilon }$ in End(V)$\mathrm{End}(V)$ such that for each B∈{A,A^?,A^ε}$B\in \{A,A^{?},A^{\epsilon }\}$ there exists a basis for V with respect to which the matrix representing B   is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. In this paper we define a family of Leonard triples said to have Racah type and classify them up to isomorphism. Moreover, we show that each of them satisfies the _Z3${\mathbb{Z}}_3$-symmetric Askey–Wilson relations. As an application, we construct all Leonard triples that have Racah type from the universal enveloping algebra U(_sl2)$U({\mathit{sl}}_2)$.     

Spin Leonard pairs

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 for each B∈{A, A*}, there exists a basis of V with respect to which the matrix representing B is diagonal and the matrix representing the other member of the pair is irreducible tridiagonal. A Leonard pair A, A* on V is said to be a spin Leonard pair whenever there exist invertible linear operators U, U* on V such that UA = A U, U*A* = A*U*, and UA* U ^?1 = U*^?1 AU*. In this case, we refer to U, U* as a Boltzmann pair for A, A*. We characterize the spin Leonard pairs. This characterization involves explicit formulas for the entries of the matrices that represent A and A* with respect to a particular basis. The formulas are expressed in terms of four algebraically independent parameters. We describe all Boltzmann pairs for a spin Leonard pair in terms of these parameters. We then describe all spin Leonard pairs associated with a given Boltzmann pair. We also describe the relationship between spin Leonard pairs and modular Leonard triples. We note a modular group action on each isomorphism class of spin Leonard pairs.     

The classification of Leonard triples of QRacah type

Let $\mathbb{K}$ denote an algebraically closed field. Let V denote a vector space over $\mathbb{K}$ with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations in $\mathrm{End}(V)$ such that for each of these transformations there exists a basis of V with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. There is a family of Leonard triples said to have QRacah type. This is the most general type of Leonard triple. We classify the Leonard triples of QRacah type up to isomorphism. We show that any Leonard triple of QRacah type satisfies the ${\mathbb{Z}}_3$-symmetric Askey-Wilson relations.     

Solvability of universal bol 2-loops

ABSTRACT

Let (A, ^?) be a structurable algebra. Then the opposite algebra (A ^op , ^?) is structurable, and we show that the triple system B ^op _A(x, y, z):=V^opx,y(z)=x(y¯z)+z(y¯x)?y(x¯z), x, y, z ∈ A, is a Kantor triple system (or generalized Jordan triple system of the second order) satisfying the condition (A). Furthermore, if A=𝔸₁?𝔸₂ denotes tensor products of composition algebras, (^?) is the standard conjugation, and (^∧) denotes a certain pseudoconjugation on A, we show that the triple systems B ^op _𝔸1?𝔸2 ( x , y¯^∧, z) are models of compact Kantor triple systems. Moreover these triple systems are simple if (dim𝔸₁, dim𝔸₂) ≠ (2, 2). In addition, we obtain an explicit formula for the canonical trace form for compact Kantor triple systems defined on tensor products of composition algebras.     

A characterization of Leonard pairs using the notion of a tail

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V that satisfy (i) and (ii) below:

(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 exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. In this paper, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.     

Normalized 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

     

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

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A^∗ : V → V that satisfy (i) and (ii) below:

(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 exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. Let X denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X with respect to the basis (ii) is tridiagonal. We show that X is spanned by

     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

q-Inverting pairs of shape 1, 2, 1

Let V denote a vector space with finite positive dimension over an algebraically closed field 𝔽. Let K, K* be a q-inverting pair on V, an ordered pair of invertible linear transformations on V satisfying certain conditions. In this article we study the q-inverting pairs with shape 1, 2, 1. We define a pair of scalars called the parameter pair for K, K*. We give six bases for V and give the action of K, K ^?1, K* and (K*)^?1 on each of these bases, respectively. We classify the q-inverting pairs of shape 1, 2, 1 in terms of the parameter pairs. We conclude with some trace formulae involving the parameter pair.     

The switching element for a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A:V→V and A^∗:V→V that satisfy (i) and (ii) below:

(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 exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. Let (resp. ) denote a basis for V referred to in (i) (resp. (ii)). We show that there exists a unique linear transformation S:V→V that sends v₀ to a scalar multiple of v_d, fixes w₀, and sends w_i to a scalar multiple of w_i for 1?i?d. We call S the switching element. We describe S from many points of view.     

Four-Dimensional Real Third-Power Associative Division Algebras

We study third-power associative division algebras A over a field 𝕂 of characteristic different from 2. Those algebras having dimension ≤2 are commutative. When 𝕂 is the field ? of real numbers, those algebras having dimension 4 are power-commutative in each of the following two cases:

1. A contains a central element;

2. A satisfies the additional identity (x, x³, x) = 0.

     

On Partial Tilting Complexes

Let R be a commutative noetherian ring and A a noetherian R-algebra. Let P ^? ∈ 𝒦^b(𝒫 _A ) with Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i > 0. We will provide a sufficient condition for P ^? to be a direct summand of a silting complex. Also, in case Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i ≠ 0, we will provide a sufficient condition for P ^? to be a direct summand of a tilting complex.     

Lower bounds on the dimension of the space of matrices transposed by a similarity transformation

Let B be a realn–by–n, invertible matrix. Sharp lower bounds are determined for the dimension of the space {A: A¹ = BAB^?1 } of real matrices A which are similar under B to their transposes A^′ . In particular, it is shown that this dimension is at least the greatest integer in (n + l)/2.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

Notes on the Kernels of Locally Finite Higher Derivations in Polynomial Rings

Let A = k^[3] be the polynomial ring in three variables over a field k, and let D be a nontrivial locally finite iterative higher derivation on A. Let A^D denote the kernel of D. In this note, we prove that, if chark > 0 and ML(A^D) ≠ A^D, then A^D ? k^[2]. As a consequence of this result, we give another proof of the cancellation theorem for k^[2] over any field k of positive characteristic.