Torsion-Type q-Deformed Heisenberg Algebra and Its Lie Polynomials

Cantuba, Rafael Reno S.; Silvestrov, Sergei

doi:10.1007/978-3-030-41850-2_24

Cited by 3 publications

(11 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We do not study ring-theoretic generalizations or finite-dimensional representation theory, but the Lie structure of associative algebras, such as the perspective in the studies [5,6,11]. We show that our results about the extended commutator algebra for the q-oscillator shed light on some related Lie structure in a q-oscillator representation of the algebra AW (3).…”

Section: 2mentioning

confidence: 94%

“…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%

“…The generators a + , a of L(H 0 ) are clearly Lie polynomials in a + , a, but then it is most natural to ask how the other basis elements of L(H 0 ) from ( 7)-( 9) can be expressed as Lie polynomials in a + , a. To show this, we shall exhibit some relations from [6,11] that express such basis elements as Lie polynomials or linear combinations of nested commutators in a + , a, but before exhibiting these relations, we first discuss a convenient notation using the adjoint map.…”

Section: Introductionmentioning

confidence: 99%

“…If q is a root of unity other than 1, there are special cases for the constructions in ( 13)- (15), and the basis (7)-(9) of L(H 0 ) is reduced according to some conditions in the exponents. These can be found in [11,Section 4.1]. If q = 0, an entirely different basis and nested commutator constructions can be found in [6,Section 4].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Cantuba¹

2023

Communications in Mathematics

View full text Add to dashboard Cite

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 $AW(3)$.

show abstract

Section: 2mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Cantuba¹

2023

Communications in Mathematics

View full text Add to dashboard Cite

show abstract

“…In all such studies [2,3,4] the focus was on the consequences of the non-Lie polynomial, deformed commutation relations on the Lie polynomials in the same algebra. 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

View full text Add to dashboard Cite

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.

show abstract

Lie structure of the Heisenberg-Weyl algebra

Cantuba

2024

International Electronic Journal of Algebra

View full text Add to dashboard Cite

As an associative algebra, the Heisenberg--Weyl algebra $\HWeyl$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and $B$ are not able to generate the whole space $\HWeyl$. We identify a non-nilpotent but solvable Lie subalgebra $\coreLie$ of $\HWeyl$, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by generators and relations. Under this presentation, we show that, for some algebra isomorphism $\isoH:\HWeyl\into\HWeyl$, the Lie algebra $\HWeyl$ is generated by the generators of $\coreLie$, together with their images under $\isoH$, and that $\HWeyl$ is the sum of $\coreLie$, $\isoH(\coreLie)$ and $\lbrak \coreLie,\isoH(\coreLie)\rbrak$.

show abstract

Torsion-Type q-Deformed Heisenberg Algebra and Its Lie Polynomials

Cited by 3 publications

References 5 publications

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

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

A Casimir Element Inexpressible as a Lie Polynomial

Lie structure of the Heisenberg-Weyl algebra

Contact Info

Product

Resources

About