Mad subalgebras of rings of differential operators on curves

Berest, Yuri; Wilson, George

doi:10.1016/j.aim.2006.09.018

Cited by 8 publications

(8 citation statements)

References 20 publications

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“…We also study the action of Aut(B) on the set of all framed mad associative subalgebras of B. We get a similar picture as in [6,9]. In addition, by presenting some concrete examples, we prove that the i (i = 2, 4, 5) are nonempty.…”

Section: Introductionmentioning

confidence: 75%

“…In [6], Berest and Wilson further determined the structure of the mad subalgebras of domains which are Morita equivalent to A 1 . For more information about mad subalgebras and their applications, we refer the reader to [6] and the references therein. In this subsection, we determine the set of all mad subalgebras of B and their framings.…”

Section: Strictly Nilpotent Elements and Strictly Semisimple Elementsmentioning

confidence: 99%

“…The maximal abelian ad-nilpotent (mad) associative subalgebras of A 1 were also determined by Dixmier in [9]. In [6], Berest and Wilson generalized Dixmier's result about mad algebras to algebras of differential operators over curves, which are domains Morita equivalent to the first Weyl algebra A 1 .…”

Section: Introductionmentioning

confidence: 98%

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Mad Subalgebras and Lie Subalgebras of an Enveloping Algebra

Tang

2010

Bull. Aust. Math. Soc.

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Let U(r(1)) denote the enveloping algebra of the two-dimensional nonabelian Lie algebra r(1) over a base field K. We study the maximal abelian ad-nilpotent (mad) associative subalgebras and finite-dimensional Lie subalgebras of U(r(1)). We first prove that the set of noncentral elements of U(r(1)) admits the Dixmier partition, U(r(1)) − K = 5 i=1 i , and establish characterization theorems for elements in i , i = 1, 3, 4. Then we determine the elements in i , i = 1, 3, and describe the eigenvalues for the inner derivation ad B x, x ∈ i , i = 3, 4. We also derive other useful results for elements in i , i = 2, 3, 4, 5.As an application, we find all framed mad subalgebras of U(r(1)) and determine all finite-dimensional nonabelian Lie algebras that can be realized as Lie subalgebras of U(r(1)). We also study the realizations of the Lie algebra r(1) in U(r(1)) in detail.2000 Mathematics subject classification: primary 17A36, 17B35; secondary 17B60.

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Section: Introductionmentioning

confidence: 75%

Section: Strictly Nilpotent Elements and Strictly Semisimple Elementsmentioning

confidence: 99%

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Mad Subalgebras and Lie Subalgebras of an Enveloping Algebra

Tang

2010

Bull. Aust. Math. Soc.

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“…In that approach, the automorphism group of an affine surface S is described via its action on a tree whose vertices correspond to certain (admissible) projective compactifications of S. Following the standard (by now) philosophy in noncommutative geometry (see, e.g., [SV]), we may think of our algebra D as the coordinate ring of a 'noncommutative affine surface'; a 'projective compactification' of D is then determined by a choice of filtration. Thus, we will define Γ by taking as its vertices a certain class of filtrations on the algebra D. It turns out that these filtrations can be naturally parametrized by an infinite-dimensional adelic Grassmannian Gr ad introduced in [W1] and studied in [W,BW,BW3] (in particular, we rely heavily on results of [BW3]). Our contruction is close in spirit to Serre's classic application of Bruhat-Tits trees for computing arithmetic subgroups of SL 2 (K) over the function fields of smooth curves (see [Se], Chap.…”

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confidence: 99%

“…For D = D(W ), the set Gr ad (D) is then a costar in Gr ad , consisting of all arrows with target at W . In [BW3], this set was denoted by Grad D .…”

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confidence: 99%

Trees, Amalgams and Calogero-Moser Spaces

Berest¹,

Eshmatov²,

Eshmatov³

2010

Preprint

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We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra A 1 . In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key rôle in our approach is played by a transitive action of the automorphism group of the free algebra C x, y on the Calogero-Moser varieties Cn defined in [BW]. Our results generalize well-known theorems of Dixmier and Makar-Limanov on automorphisms of A 1 , answering an old question of Stafford (see [S]). Finally, we propose a natural extension of the Dixmier Conjecture for A 1 to the class of Morita equivalent algebras.

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Multitransitivity of Calogero-Moser Spaces

Berest

Eshmatov

2015

Transformation Groups

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Let G be the group of unimodular automorphisms of a free associative Calgebra on two generators. A theorem of G. Wilson and the first author [BW] asserts that the natural action of G on the Calogero-Moser spaces Cn is transitive for all n ∈ N. We extend this result in two ways: first, we prove that the action of G on Cn is doubly transitive, meaning that G acts transitively on the configuration space of ordered pairs of distinct points in Cn; second, we prove that the diagonal action of G on Cn 1 × Cn 2 × · · · × Cn m is transitive provided n 1 , n 2 , . . . , nm are pairwise distinct numbers.

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Mad subalgebras of rings of differential operators on curves

Cited by 8 publications

References 20 publications

Mad Subalgebras and Lie Subalgebras of an Enveloping Algebra

Mad Subalgebras and Lie Subalgebras of an Enveloping Algebra

Trees, Amalgams and Calogero-Moser Spaces

Multitransitivity of Calogero-Moser Spaces

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