Lie Polynomial Characterization Problems

Cantuba, Rafael Reno S.; Silvestrov, Sergei

doi:10.1007/978-3-030-41850-2_25

Cited by 5 publications

(7 citation statements)

References 25 publications

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“…The universal Askey-Wilson algebra also intersects the theory of free Lie algebras, see e.g. [64,65]. The evaluated Askey-Wilson algebra Z q (m 1 , m 2 , m 3 ) is the quotient of aw(3) by the supplementary relations…”

Section: A Jungle Of Askey-wilson Algebrasmentioning

confidence: 99%

The Askey–Wilson algebra and its avatars

Crampé¹,

Frappat²,

Gaboriaud³

et al. 2021

J. Phys. A: Math. Theor.

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The original Askey–Wilson algebra introduced by Zhedanov encodes the bispectrality properties of the eponym polynomials. The name Askey–Wilson algebra is currently used to refer to a variety of related structures that appear in a large number of contexts. We review these versions, sort them out and establish the relations between them. We focus on two specific avatars. The first is a quotient of the original Zhedanov algebra; it is shown to be invariant under the Weyl group of type D 4 and to have a reflection algebra presentation. The second is a universal analogue of the first one; it is isomorphic to the Kauffman bracket skein algebra (KBSA) of the four-punctured sphere and to a subalgebra of the universal double affine Hecke algebra ( C 1 ∨ , C 1 ) . This second algebra emerges from the Racah problem of U q ( s l 2 ) and is related via an injective homomorphism to the centralizer of U q ( s l 2 ) in its threefold tensor product. How the Artin braid group acts on the incarnations of this second avatar through conjugation by R-matrices (in the Racah problem) or half Dehn twists (in the diagrammatic KBSA picture) is also highlighted. Attempts at defining higher rank Askey–Wilson algebras are briefly discussed and summarized in a diagrammatic fashion.

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Section: A Jungle Of Askey-wilson Algebrasmentioning

confidence: 99%

The Askey–Wilson algebra and its avatars

Crampé¹,

Frappat²,

Gaboriaud³

et al. 2021

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…Proof. We first prove (7). For the case m, n ∈ Z + , using the first relation in (2) and double induction on m, n it is routine to show that…”

Section: Reordering Formula In a Qmentioning

confidence: 99%

“…The studies [2,3,4] have motivated further progress as reported in [5,6], in which central extensions and torsion-type deformation parameters were considered, respectively. A survey of the particular type of Lie structure, in associative algebras being described here, can be found in [7].…”

Section: Introductionmentioning

confidence: 99%

A Casimir Element Inexpressible as a Lie Polynomial

Cantuba

2021

International Electronic Journal of Algebra

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Let q be a scalar that is not a root of unity. We show that any nonzero polynomial in the Casimir element of the Fairlie-Odesskii algebra U q (so 3 ) cannot be expressed in terms of only Lie algebra operations performed on the generators I 1 , I 2 , I 3 in the usual presentation of U q (so 3 ). Hence, the vector space sum of the center of U q (so 3 ) and the Lie subalgebra of U q (so 3 ) generated by I 1 , I 2 , I 3 is direct.

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“…Some works published after [16] continue to refer to F ⟨A, B⟩ /(AB − qBA − 1) as H(q) or as the q-deformed Heisenberg algebra. Said works come from varied fields of mathematics, such as Ring Theory [15,17], Lie algebras [6,8,9,10,11], Mathematical Physics [7,19], and algebraic curves [13]. All these, and most probably many more, refer to H(q) = F ⟨A, B⟩ /(AB − qBA − 1) as the q-deformed Heisenberg algebra.…”

Section: Introductionmentioning

confidence: 99%

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Cantuba¹

2023

Communications in Mathematics

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Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.

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Lie Polynomial Characterization Problems

Cited by 5 publications

References 25 publications

The Askey–Wilson algebra and its avatars

The Askey–Wilson algebra and its avatars

A Casimir Element Inexpressible as a Lie Polynomial

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

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