The graph of a Weyl algebra endomorphism

2020 ◽  
Author(s):  
Niels Lauritzen ◽  
Jesper Funch Thomsen
Keyword(s):  
Weyl Algebra ◽  
Algebra Endomorphism

scholarly journals On the eigenvector algebra of the product of elements with commutator one in the first Weyl algebra

2011 ◽  
Vol 151 (2) ◽  
pp. 245-262
Author(s):  
V. V. BAVULA
Keyword(s):  
Characteristic Zero ◽  
Weyl Algebra ◽  
Algebra Homomorphism ◽  
Inner Derivation ◽  
Algebra Over A Field ◽  
Algebra Endomorphism

Let A1 = K〈X, Y|[Y, X]=1〉 be the (first) Weyl algebra over a field K of characteristic zero. It is known that the set of eigenvalues of the inner derivation ad(YX) of A1 is ℤ. Let A1 → A1, X ↦ x, Y ↦ y, be a K-algebra homomorphism, i.e. [y, x] = 1. It is proved that the set of eigenvalues of the inner derivation ad(yx) of the Weyl algebra A1 is ℤ and the eigenvector algebra of ad(yx) is K〈x, y〉 (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: is an algebra endomorphism of A1 an automorphism?).

scholarly journals About Dixmier's conjecture

2015 ◽  
Vol 14 (10) ◽  
pp. 1550140
Author(s):  
Vered Moskowicz
Keyword(s):  
Conjugacy Class ◽  
Characteristic Zero ◽  
Weyl Algebra ◽  
Zero Field ◽  
Algebra Endomorphism

The well-known Dixmier conjecture [5] asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism. We bring a hopefully easier to solve conjecture, called the γ, δ conjecture, and show that it is equivalent to the Dixmier conjecture. In the group generated by automorphisms and anti-automorphisms of A1, all the involutions belong to one conjugacy class, hence: • Every involutive endomorphism from (A1, γ) to (A1, δ) is an automorphism (γ and δ are two involutions on A1). • Given an endomorphism f of A1 (not necessarily an involutive endomorphism), if one of f(X), f(Y) is symmetric or skew-symmetric (with respect to any involution on A1), then f is an automorphism.

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 Algebra endomorphisms and derivations of some localized down-up algebras

2014 ◽  
Vol 14 (03) ◽  
pp. 1550034 ◽  
Author(s):  
Xin Tang
Keyword(s):  
Automorphism Group ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Root Of Unity ◽  
Hochschild Cohomology Group ◽  
Algebra Endomorphism

We study algebra endomorphisms and derivations of some localized down-up algebras A𝕊(r + s, -rs). First, we determine all the algebra endomorphisms of A𝕊(r + s, -rs) under some conditions on r and s. We show that each algebra endomorphism of A𝕊(r + s, -rs) is an algebra automorphism if rmsn = 1 implies m = n = 0. When r = s-1 = q is not a root of unity, we give a criterion for an algebra endomorphism of A𝕊(r + s, -rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A𝕊(r + s, -rs). We also show that each surjective algebra endomorphism of the down-up algebra A(r + s, -rs) is an algebra automorphism in either case. Second, we determine all the derivations of A𝕊(r + s, -rs) and calculate its first degree Hochschild cohomology group.

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
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