Maximal Abelian Subalgebra Research Articles

New classes of quadratically integrable systems in magnetic fields: The generalized cylindrical and spherical cases

We study integrable and superintegrable systems with magnetic field possessing quadratic integrals of motion on the three-dimensional Euclidean space. In contrast with the case without vector potential, the corresponding integrals may no longer be connected to separation of variables in the Hamilton–Jacobi equation and can have more general leading order terms. We focus on two cases extending the physically relevant cylindrical- and spherical-type integrals. We find three new integrable systems in the generalized cylindrical case but none in the spherical one. We conjecture that this is related to the presence, respectively absence, of maximal abelian Lie subalgebra of the three-dimensional Euclidean algebra generated by first order integrals in the limit of vanishing magnetic field. By investigating superintegrability, we find only one (minimally) superintegrable system among the integrable ones. It does not separate in any orthogonal coordinate system. This system provides a mathematical model of a helical undulator placed in an infinite solenoid.

Non-relativistic string monodromies

Spectral curve methods proved to be powerful techniques in the context of relativistic integrable string theories, since they allow us to derive the semiclassical spectrum from the minimal knowledge of a Lax pair and a classical string solution. In this paper we initiate the study of the spectral curve for non-relativistic strings in AdS5 × S5. First, we show that for string solutions whose Lax connection is independent of σ, the eigenvalues of the monodromy matrix do not have any spectral parameter dependence. We remark that this particular behaviour also appears for relativistic strings in flat space. Second, for some simple non-relativistic string solutions where the path ordered exponential of the Lax connection can be computed, we show that the monodromy matrix is either diagonalisable with quasi-momenta independent of the spectral parameter, or non-diagonalisable. For the latter case, we propose a notion of generalised quasi-momenta, based on maximal abelian subalgebras, which retain a dependence on the spectral parameter.

Abelian subalgebras and ideals of maximal dimension in Zinbiel algebras

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension 1 and supersolvable Zinbiel algebras in which such subalgebras have codimension 2, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for α and β for the low dimensional Zinbiel algebras over the complex field that have been classified. Communicated by Alberto Elduque

Structure of extensions of free Araki–Woods factors

We investigate the structure of crossed product von Neumann algebras arising from Bogoljubov actions of countable groups on Shlyakhtenko’s free Araki–Woods factors. Among other results, we settle the questions of factoriality and Connes’ type classification. We moreover provide general criteria regarding fullness and strong solidity. As an application of our main results, we obtain examples of type III 0 factors that are prime, have no Cartan subalgebra and possess a maximal amenable abelian subalgebra. We also obtain a new class of strongly solid type III factors with prescribed Connes’ invariants that are not isomorphic to any free Araki–Woods factors.

Maximal abelian subalgebras of Banach algebras

Let A be a commutative, unital Banach algebra. We consider the number of different non-commutative, unital Banach algebras C such that A is a maximal abelian subalgebra in C. For example, we shall prove that, in the case where A is an infinite-dimensional, unital Banach function algebra, A is a maximal abelian subalgebra in infinitely-many closed subalgebras of B ( A ) such that no two distinct subalgebras are isomorphic; the same result holds for certain examples A that are local algebras. We shall also give examples of uniform algebras of the form C ( K ) , where K is a compact space, with the property that there exists a family of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras C such that each C contains B ( C ( K ) ) as a closed subalgebra and is such that C ( K ) is a maximal abelian subalgebra in C.

Diagonalizability of Quantum Markov States on Trees

We introduce quantum Markov states (QMS) in a general tree graph $$G= (V, E)$$ , extending the Cayley tree’s case. We investigate the Markov property w.r.t. the finer structure of the considered tree. The main result of the present paper concerns the diagonalizability of a locally faithful QMS $$\varphi $$ on a UHF-algebra $${\mathcal {A}}_V$$ over the considered tree by means of a suitable conditional expectation into a maximal abelian subalgebra. Namely, we prove the existence of a Umegaki conditional expectation $${\mathfrak {E}} : {\mathcal {A}}_V \rightarrow {\mathcal {D}}_V$$ such that $$\varphi =\varphi _{\lceil {\mathcal {D}}_V}\circ {\mathfrak {E}}$$ . Moreover, it is clarified the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra $${\mathcal {D}}_V$$ .

Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere

Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is also the case for the construction of their related symmetry algebra which take usually the form of a finitely generated quadratic algebra. These algebras often display structure constants which depend on the central elements and in particular on the Hamiltonian. In this paper, we develop a new approach reexamining the quadratically superintegrable system on the two-sphere for which a symmetry algebra is known to be the Racah algebra R(3). Such a model is related to the 59 two dimensional quadratically superintegrable systems on conformally flat spaces via contractions and limits. We demonstrate that using further polynomials of degree 2, 3 and 4 in the enveloping algebra of su(3) one can generate an algebra based only on abstract commutation relations of su(3) Lie algebra without explicit constraints on the representations or realisations. This construction relies on the maximal Abelian subalgebra, also called MASA, which are the Cartan generators and their commutant. We obtain a new six-dimensional cubic algebra where the structure constant are integer numbers which reduce from a quartic algebra for which the structure constant depend on the Cartan generator and the Casimir invariant. We also present other form of the symmetry algebra using the quadratic and cubic Casimir invariants of su(3). It reduces as the known quadratic Racah algebra R(3) only when using an explicit realization. This algebraic structure describes the symmetry of the quadratically superintegrable systems on the 2 sphere. We also present a contraction to another six-dimensional cubic algebra which would corresponding to the symmetry algebra of a Smorodinsky–Winternitz model.

Connes' bicentralizer problem for q‐deformed Araki–Woods algebras

Let $(H_{\mathbf{R}}, U_t)$ be any strongly continuous orthogonal representation of $\mathbf{R}$ on a real (separable) Hilbert space $H_{\mathbf{R}}$. For any $q\in (-1,1)$, we denote by $\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ the $q$-deformed Araki-Woods algebra introduced by Shlyakhtenko and Hiai. In this paper, we prove that $\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ has trivial bicentralizer if it is a type $\rm III_1$ factor. In particular, we obtain that $\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ always admits a maximal abelian subalgebra that is the range of a faithful normal conditional expectation. Moreover, using Sniady's work, we derive that $\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ is a full factor provided that the weakly mixing part of $(H_{\mathbf{R}}, U_t)$ is nonzero.

On split involutive regular BiHom-Lie superalgebras

Abstract The goal of this paper is to examine the structure of split involutive regular BiHom-Lie superalgebras, which can be viewed as the natural generalization of split involutive regular Hom-Lie algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular BiHom-Lie superalgebra L {\mathfrak{L}} is of the form L = U + ∑ α I α {\mathfrak{L}}=U+{\sum }_{\alpha }{I}_{\alpha } with U a subspace of a maximal abelian subalgebra H and any I α , a well-described ideal of L {\mathfrak{L}} , satisfying [I α , I β ] = 0 if [α] ≠ [β]. In the case of L {\mathfrak{L}} being of maximal length, the simplicity of L {\mathfrak{L}} is also characterized in terms of connections of roots.

On split regular BiHom-Leibniz superalgebras

The goal of this paper is to study the structure of split regular BiHom-Leiniz superalgebras, which is a natural generalization of split regular Hom-Leiniz algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Leiniz superalgebras $\mathfrak{L}$ is of the form $\mathfrak{L}=U+\sum_{\a}I_\a$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\a}$, a well described ideal of $\mathfrak{L}$, satisfying $[I_\a, I_\b]= 0$ if $[\a]\neq [\b]$. In the case of $\mathfrak{L}$ being of maximal length, the simplicity of $\mathfrak{L}$ is also characterized in terms of connections of roots.

Thin II1 factors with no Cartan subalgebras

It is a wide open problem to give an intrinsic criterion for a II_1 factor $M$ to admit a Cartan subalgebra $A$. When $A \subset M$ is a Cartan subalgebra, the $A$-bimodule $L^2(M)$ is "simple" in the sense that the left and right action of $A$ generate a maximal abelian subalgebra of $B(L^2(M))$. A II_1 factor $M$ that admits such a subalgebra $A$ is said to be s-thin. Very recently, Popa discovered an intrinsic local criterion for a II_1 factor $M$ to be s-thin and left open the question whether all s-thin II_1 factors admit a Cartan subalgebra. We answer this question negatively by constructing s-thin II_1 factors without Cartan subalgebras.

On masas in q-deformed von Neumann algebras

We study certain $q$-deformed analogues of the maximal abelian subalgebras of the group von Neumann algebras of free groups. The radial subalgebra is defined for Hecke deformed von Neumann algebras of the Coxeter group $(\mathbb{Z}/{2\mathbb{Z}})^{\star k}$ and shown to be a maximal abelian subalgebra which is singular and with Puk\'anszky invariant $\{\infty\}$. Further all non-equal generator masas in the $q$-deformed Gaussian von Neumann algebras are shown to be mutually non-unitarily conjugate.

Constructing MASAs with prescribed properties

We consider an iterative procedure for constructing maximal abelian ∗-subalgebras (MASAs) satisfying prescribed properties in II1 factors. This method pairs well with the intertwining by bimodules technique and with properties of the MASA and of the ambient factor that can be described locally. We obtain such a local characterization for II1 factors M that have an s-MASA, A⊂M (i.e., for which A∨JAJ is maximal abelian in B(L2M)), and use this strategy to prove that any factor in this class has uncountably many nonintertwinable singular (resp., semiregular) s-MASAs.

The dual pair [formula omitted], the Dirac equation and the Bannai–Ito algebra

The Bannai–Ito algebra can be defined as the centralizer of the coproduct embedding of osp(1|2) in osp(1|2)⊗n. It will be shown that it is also the commutant of a maximal Abelian subalgebra of o(2n) in a spinorial representation and an embedding of the Racah algebra in this commutant will emerge. The connection between the two pictures for the Bannai–Ito algebra will be traced to the Howe duality which is embodied in the Pin(2n)×osp(1|2) symmetry of the massless Dirac equation in R2n. Dimensional reduction to Rn will provide an alternative to the Dirac–Dunkl equation as a model with Bannai–Ito symmetry.

On a Poisson Summation Formula for Noncommutative Tori

It is proved that a maximal abelian subalgebra of the noncommutative torus commutes with the Laplace operator on a complex torus. As a corollary, one gets an analog of the Poisson summation formula for noncommutative tori.

On Boolean ranges of Banaschewski functions

We construct a countable lattice $${\varvec{\mathcal {S}}}$$ isomorphic to a bounded sublattice of the subspace lattice of a vector space with two non-iso-morphic maximal Boolean sublattices. We represent one of them as the range of a Banaschewski function and we prove that this is not the case of the other. Hereby we solve a problem of F. Wehrung. We study coordinatizability of the lattice $${\varvec{\mathcal {S}}}$$ . We prove that although it does not contain a 3-frame, the lattice $${\varvec{\mathcal {S}}}$$ is coordinatizable. We show that the two maximal Boolean sublattices correspond to maximal Abelian regular subalgebras of the coordinatizating ring.

On split regular Hom-Lie superalgebras

We introduce the class of split regular Hom-Lie superalgebras as the natural extension of the one of split Hom-Lie algebras and Lie superalgebras, and study its structure by showing that an arbitrary split regular Hom-Lie superalgebra L is of the form L = U + ∑ j I j with U a linear subspace of a maximal abelian graded subalgebra H and any I j a well described (split) ideal of L satisfying [ I j , I k ] = 0 if j ≠ k . Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its simple ideals.

Symmetry algebra for the generic superintegrable system on the sphere

The goal of the present paper is to provide a detailed study of irreducible representations of the algebra generated by the symmetries of the generic quantum superintegrable system on the d-sphere. Appropriately normalized, the symmetry operators preserve the space of polynomials. Under mild conditions on the free parameters, maximal abelian subalgebras of the symmetry algebra, generated by Jucys-Murphy elements, have unique common eigenfunctions consisting of families of Jacobi polynomials in d variables. We describe the action of the symmetries on the basis of Jacobi polynomials in terms of multivariable Racah operators, and combine this with different embeddings of symmetry algebras of lower dimensions to prove that the representations restricted on the space of polynomials of a fixed total degree are irreducible.

The inner structure of boundary quotients of right LCM semigroups

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,\theta)$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in $2 \leq p<\infty$ generators to the $p$-adic ring $C^*$-algebra. For the case where $P$ is abelian and $C^*(G)$ is a maximal abelian subalgebra, we establish a picture for the automorphisms of the boundary quotient that fix $C^*(G)$ pointwise. This allows us to show that they form a maximal abelian subgroup of the entire automorphism group. The picture also leads to the surprising outcome that, for integral dynamics, every automorphism that fixes one of the natural Cuntz subalgebras pointwise is necessarily a gauge automorphism. Many of the automorphisms we consider are shown to be outer.

Maximal abelian subalgebras of C⁎-algebras associated with complex dynamical systems and self-similar maps

We consider an analogy among Markov shifts, complex dynamical systems and self-similar maps. Their dynamics are given by 0–1 matrices A, rational functions R and self-similar maps γ on a compact metric space K, respectively. If the 0–1 matrix A is irreducible and not a permutation, then the Cuntz–Krieger algebra OA is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set JR and the self-similar map γ satisfies the open set condition respectively, then the associated C⁎-algebras OR(JR) and Oγ(K) are simple and purely infinite. Let ΣA be the associated infinite path space for the 0–1 matrix A, then C(ΣA) is known to be a maximal abelian subalgebra of OA. In this paper we shall show that C(JR) is a maximal abelian subalgebra of OR(JR) and C(K) is a maximal abelian subalgebra of Oγ(K).