Cyclic irreducible non-holonomic modules over the Weyl algebra: An algorithmic characterization

Bernstein's famous result, that any non-zero module M over the n-th Weyl algebra An satisfies GKdim(M)≥GKdim(An)/2, does not carry over to arbitrary simple affine algebras, as is shown by an example of McConnell. Bavula introduced the notion of filter dimension of simple algebra to explain this failure. Here, we introduce the faithful dimension of a module, a variant of the filter dimension, to investigate this phenomenon further and to study a revised definition of holonomic modules. We compute the faithful dimension for certain modules over a variant of the McConnell example to illustrate the utility of this new dimension.

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Simple Holonomic Modules over the Second Weyl Algebra A2

Advances in Mathematics ◽

10.1006/aima.1999.1835 ◽

2000 ◽

Vol 150 (1) ◽

pp. 80-116 ◽

Cited By ~ 8

Author(s):

V. Bavula ◽

F. van Oystaeyen

Keyword(s):

Weyl Algebra ◽

Holonomic Modules

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

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202105059 ◽

2002 ◽

Vol 31 (9) ◽

pp. 513-553 ◽

Cited By ~ 3

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.

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Cohen–Macaulay Modules and Holonomic Modules Over Filtered Rings

Communications in Algebra ◽

10.1080/00927870802248605 ◽

2009 ◽

Vol 37 (2) ◽

pp. 406-430

Author(s):

Hiroki Miyahara ◽

Kenji Nishida

Keyword(s):

Holonomic Modules

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Path decompositions of digraphs and their applications to Weyl algebra

Advances in Applied Mathematics ◽

10.1016/j.aam.2015.03.005 ◽

2015 ◽

Vol 67 ◽

pp. 36-54 ◽

Cited By ~ 3

Author(s):

Askar Dzhumadil'daev ◽

Damir Yeliussizov

Keyword(s):

Weyl Algebra ◽

Path Decompositions

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Specht polynomials and modules over the Weyl algebra

Afrika Matematika ◽

10.1007/s13370-018-0642-9 ◽

2018 ◽

Vol 30 (1-2) ◽

pp. 279-290

Author(s):

Ibrahim Nonkané

Keyword(s):

Weyl Algebra

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The classical limit of a state on the Weyl algebra

Journal of Mathematical Physics ◽

10.1063/1.5013249 ◽

2018 ◽

Vol 59 (11) ◽

pp. 112102 ◽

Cited By ~ 3

Author(s):

Benjamin H. Feintzeig

Keyword(s):

Classical Limit ◽

Weyl Algebra

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

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091514000303 ◽

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:

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