The q-Deformed Weyl Algebra and the Meromorphic Weyl Algebra

2015 ◽  
pp. 249-300
Keyword(s):  
Weyl Algebra

scholarly journals Quantum relativistic Toda chain at root of unity: isospectrality, modifiedQ-operator, and functional Bethe ansatz

2002 ◽  
Vol 31 (9) ◽  
pp. 513-553 ◽  
Author(s):  
Stanislav Pakuliak ◽  
Sergei Sergeev
Keyword(s):  
Bethe Ansatz ◽  
Weyl Algebra ◽  
Separation Of Variables ◽  
Toda Chain ◽  
Special Choice ◽  
Finite Dimensional Representation ◽  
Classical Counterpart ◽  
Discrete Integrable System ◽  
Root Of Unity ◽  
Finite Dimensional

We investigate anN-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra withqbeingNth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter'sQ-operators. The classical counterpart of the modifiedQ-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modifiedQ-operators.

scholarly journals Path decompositions of digraphs and their applications to Weyl algebra

2015 ◽  
Vol 67 ◽  
pp. 36-54 ◽  
Author(s):  
Askar Dzhumadil'daev ◽  
Damir Yeliussizov
Keyword(s):  
Weyl Algebra ◽  
Path Decompositions

Specht polynomials and modules over the Weyl algebra

2018 ◽  
Vol 30 (1-2) ◽  
pp. 279-290
Author(s):  
Ibrahim Nonkané
Keyword(s):  
Weyl Algebra

scholarly journals The classical limit of a state on the Weyl algebra

2018 ◽  
Vol 59 (11) ◽  
pp. 112102 ◽  
Author(s):  
Benjamin H. Feintzeig
Keyword(s):  
Classical Limit ◽  
Weyl Algebra

scholarly journals The Group of Automorphisms of the Algebra of one-Sided Inverses of a Polynomial Algebra. II

2015 ◽  
Vol 58 (3) ◽  
pp. 543-580
Author(s):  
V. V. Bavula
Keyword(s):  
Weyl Algebra ◽  
Differential Operators ◽  
Polynomial Algebra ◽  
Group Of Automorphisms ◽  
Algebra Ii ◽  
Noetherian Property

AbstractThe algebra of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra is isomorphic to the algebra of scalar integro-differential operators provided that char(K) = 0. Ignoring the non-Noetherian property, the algebra belongs to a family of algebras like the nth Weyl algebra An and the polynomial algebra P2n. Explicit generators are found for the group Gn of automorphisms of the algebra and for the group of units of (both groups are huge). An analogue of the Jacobian homomorphism AutK-alg (Pn) → K* is introduced for the group Gn (notice that the algebra is non-commutative and neither left nor right Noetherian). The polynomial Jacobian homomorphism is unique. Its analogue is also unique for n > 2 but for n = 1, 2 there are exactly two of them. The proof is based on the following theorem that is proved in the paper:

A strict Positivstellensatz for the Weyl algebra

2005 ◽  
Vol 331 (4) ◽  
pp. 779-794 ◽  
Author(s):  
Konrad Schm�dgen
Keyword(s):  
Weyl Algebra

scholarly journals The Weyl algebra, spherical harmonics, and Hahn polynomials

2002 ◽  
Vol 55 ◽  
pp. 309-322
Author(s):  
Ewa Gnatowska ◽  
Aleksander Strasburger
Keyword(s):  
Spherical Harmonics ◽  
Weyl Algebra ◽  
Hahn Polynomials

scholarly journals On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

10.37236/6699 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Yu-Chang Liang ◽  
Tsai-Lien Wong
Keyword(s):  
Weyl Algebra ◽  
Stirling Numbers ◽  
Stirling Number ◽  
Combinatorial Structures ◽  
Normal Ordering ◽  
Rook Placements ◽  
Threshold Graphs ◽  
Combinatorial Interpretations ◽  
Ferrers Boards

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.

scholarly journals Centralizers of Rank One in the First Weyl Algebra

2021 ◽  
Author(s):  
Leonid Makar-Limanov ◽  
Keyword(s):  
Weyl Algebra ◽  
Genus Zero ◽  
Rank One

Centralizers of rank one in the first Weyl algebra have genus zero.

scholarly journals On the Automorphism Group of the First Weyl Algebra

2013 ◽  
Vol 42 (1) ◽  
pp. 81-95 ◽  
Author(s):  
M. K. Kouakou ◽  
A. Tchoudjem
Keyword(s):  
Automorphism Group ◽  
Weyl Algebra
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