scholarly journals Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups

10.37236/5264 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Silvia Goodenough ◽  
Christian Lavault
Keyword(s):  
Weyl Algebra ◽  
Differential Operators ◽  
Combinatorial Structures ◽  
Fock Representation ◽  
Riordan Arrays ◽  
Riordan Group

In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg—Weyl, its Bargmann—Fock representation with differential operators and the associated one-parameter group.Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e., substitutions with prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the "striped" Riordan subgroups; further, a striped quasigroup and a semigroup are also examined. A few  applications to combinatorial structures are also briefly addressed in the Appendix.

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:

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 On ζn-Weyl algebra Wr(ζn, Z)

1989 ◽  
Vol 113 ◽  
pp. 153-159 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Associative Algebra ◽  
Weyl Algebra ◽  
Differential Operators

Weyl algebra is an associative algebra generated by two elements â and a over R such that the generating relation is given byâa — aâ = 1,which is isomorphic to the algebra of differential operators

DIXMIER'S PROBLEM 6 FOR SOMEWHAT COMMUTATIVE ALGEBRAS AND DIXMIER'S PROBLEM 3 FOR THE RING OF DIFFERENTIAL OPERATORS ON A SMOOTH IRREDUCIBLE AFFINE CURVE

2005 ◽  
Vol 04 (05) ◽  
pp. 577-586 ◽  
Author(s):  
V. V. BAVULA
Keyword(s):  
Weyl Algebra ◽  
Differential Operators ◽  
Short Proof ◽  
Prime Factor ◽  
Affirmative Answer ◽  
Inner Derivation ◽  
Enveloping Algebra ◽  
Commutative Algebras ◽  
Rings Of Differential Operators ◽  
Kirillov Dimension

In [6], J. Dixmier posed six problems for the Weyl algebra A1 over a field K of characteristic zero. Problems 3, 5,and 6 were solved respectively by Joseph and Stein [7]; the author [1]; and Joseph [7]. Problems 1, 2, and 4 are still open. For an arbitrary algebra A, Dixmier's problem 6 is essentially aquestion: whether an inner derivation of the algebra A of the type ad f(a), a ∈ A, f(t) ∈ K[t], deg t(f(t)) > 1, has a nonzero eigenvalue. We prove that the answer is negative for many classes of algebras (e.g., rings of differential operators [Formula: see text] on smooth irreducible algebraic varieties, all prime factor algebras of the universal enveloping algebra [Formula: see text] of a completely solvable algebraic Lie algebra [Formula: see text]). This gives an affirmative answer (with a short proof) to an analogue of Dixmier's Problem 6 for certain algebras of small Gelfand–Kirillov dimension, e.g. the ring of differential operators [Formula: see text] on a smooth irreducible affine curve X, Usl(2), etc. (see [3] for details). In this paper an affirmative answer is given to an analogue of Dixmier's Problem 3 but for the ring [Formula: see text].

scholarly journals THE GROUP OF AUTOMORPHISMS OF THE FIRST WEYL ALGEBRA IN PRIME CHARACTERISTIC AND THE RESTRICTION MAP

2009 ◽  
Vol 51 (2) ◽  
pp. 263-274 ◽  
Author(s):  
V. V. BAVULA
Keyword(s):  
Explicit Formula ◽  
Weyl Algebra ◽  
Differential Operators ◽  
Algebraic Groups ◽  
Restriction Map ◽  
Prime Characteristic ◽  
Group Of Automorphisms ◽  
Infinite Dimensional ◽  
Image Position

AbstractLet K be a perfect field of characteristic p > 0; A1 := K〈x, ∂|∂x−x∂=1〉 be the first Weyl algebra; and Z:=K[X:=xp, Y:=∂p] be its centre. It is proved that (i) the restriction map res : AutK(A1)→ AutK(Z), σ ↦ σ|Z is a monomorphism with im(res) = Γ := {τ ∈ AutK(Z)|(τ)=1}, where (τ) is the Jacobian of τ, (Note that AutK(Z)=K* ⋉ Γ, and if K is not perfect then im(res) ≠ Γ.); (ii) the bijection res : AutK(A1) → Γ is a monomorphism of infinite dimensional algebraic groups which is not an isomorphism (even if K is algebraically closed); (iii) an explicit formula for res−1 is found via differential operators (Z) on Z and negative powers of the Fronenius map F. Proofs are based on the following (non-obvious) equality proved in the paper:

scholarly journals Differential operators and Higher Specht polynomials

2021 ◽  
Vol 2090 (1) ◽  
pp. 012096
Author(s):  
Ibrahim Nonkané ◽  
Léonard Todjihounde
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Weyl Algebra ◽  
Differential Operators ◽  
Symmetric Groups ◽  
Symmetric Polynomial ◽  
Symmetric Polynomials ◽  
Polynomial Representation ◽  
Invariant Differential ◽  
Moser System

Abstract In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

scholarly journals Differential operators and reection group of type Dn

2021 ◽  
Vol 2090 (1) ◽  
pp. 012098
Author(s):  
Ibrahim Nonkané ◽  
Latévi M. Lawson
Keyword(s):  
Polynomial Ring ◽  
Weyl Algebra ◽  
Differential Operators ◽  
Invariant Polynomials ◽  
Invariant Differential Operators ◽  
Differential Structure ◽  
Invariant Differential

Abstract In this note, we study the actions of rational quantum Olshanetsky-Perelomov systems for finite reflections groups of type D n . we endowed the polynomial ring C[x 1,..., xn ] with a differential structure by using directly the action of the Weyl algebra associated with the ring C[x 1,..., xn ] W of invariant polynomials under the reflections groups W after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the reflections groups. We use the higher Specht polynomials associated with the representation of the reflections group W to exhibit the generators of its simple components.

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