Maximal Commutative Subalgebras Research Articles

Realisations of Racah algebras using Jacobi operators and convolution identities

Using the representation theory of sl2 and an appropriate model for tensor product of lowest weight Verma modules, we give a realisation first of the Hahn algebra, and then of the Racah algebra, using Jacobi differential operators. While doing so we recover some known convolution formulas for Jacobi polynomials involving Hahn and Racah polynomials. Similarly, we produce realisations of the higher rank Racah algebras in which maximal commutative subalgebras are realised in terms of Jacobi differential operators.

Spectrality in Convex Sequential Effect Algebras

For convex and sequential effect algebras, we study spectrality in the sense of Foulis. We show that under additional conditions (strong archimedeanity, closedness in norm and a certain monotonicity property of the sequential product), such effect algebra is spectral if and only if every maximal commutative subalgebra is monotone sigma -complete. Two previous results on existence of spectral resolutions in this setting are shown to require stronger assumptions.

Maximal commutative subalgebras of Leavitt path algebras

Let [Formula: see text] be a field, and let [Formula: see text] be a row-finite (directed) graph. We present a construction of a wealth of maximal commutative subalgebras of the Leavitt path algebra [Formula: see text], which is a far-reaching generalization of the construction of the commutative core as a maximal commutative subalgebra of [Formula: see text].

Deformed Calogero–Moser Operators and Ideals of Rational Cherednik Algebras

We introduce a class of hyperplane arrangements $$\mathcal {A}$$ in $${\mathbb {C}}^n$$ that generalise the locus configurations of Chalykh, Feigin and Veselov. To such an arrangement we associate a second order partial differential operator of Calogero–Moser type and prove that this operator is completely integrable (in the sense that its centraliser in $$\mathcal {D}({\mathbb {C}}^n\!\setminus \!\mathcal {A})$$ contains a maximal commutative subalgebra of Krull dimension n). Our approach is based on the study of shift operators and associated ideals in spherical Cherednik algebras that may be of independent interest. Examples include all known completely integrable deformations of Calogero–Moser operators with rational potentials. In addition, we construct new families of examples, including a BC-type generalisation of the deformed Calogero-Moser operators recently found by Gaiotto and Rapčák. We describe these examples in a unified representation-theoretic framework of rational Cherednik algebras.

Faà di Bruno Hopf algebras

This is a short review on the Faà di Bruno formulas, implementing composition of real-analytic functions, and a Hopf algebra associated to such formulas. This structure provides, among several other things, a short proof of the Lie-Scheffers theorem, and relates the Lagrange inversion formulas with antipodes. It is also the maximal commutative Hopf subalgebra of the one used by Connes and Moscovici to study diffeomorphisms in a noncommutative geometry setting. The link of Faà di Bruno formulas with the theory of set partitions is developed in some detail.

Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras

Abstract Considering prime Leavitt path algebras L K ⁢ ( E ) {L_{K}(E)} , with E being an arbitrary graph with at least two vertices, and K being any field, we construct a class of maximal commutative subalgebras of L K ⁢ ( E ) {L_{K}(E)} such that, for every algebra A from this class, A has zero intersection with the commutative core ℳ K ⁢ ( E ) {\mathcal{M}_{K}(E)} of L K ⁢ ( E ) {L_{K}(E)} defined and studied in [C. Gil Canto and A. Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 2018, 227–248]. We also give a new proof of the maximality, as a commutative subalgebra, of the commutative core ℳ R ⁢ ( E ) {\mathcal{M}_{R}(E)} of an arbitrary Leavitt path algebra L R ⁢ ( E ) {L_{R}(E)} , where E is an arbitrary graph and R is a commutative unital ring.

Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology

We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents in the category of Soergel bimodules, and they correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. We define a family of differentials dN on these dg algebras and conjecture that their homology matches that of the glN projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende.

Commuting Ordinary Differential Operators and the Dixmier Test

The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial coefficients, while the classification of commutative subalgebras of the Weyl algebra is in itself an important open problem. Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator $L$ in normal form, following the approach of K.R. Goodearl, with the ultimate goal of obtaining such bases by computational routines. Our first step is to establish the Dixmier test, based on a lemma by J. Dixmier and the choice of a suitable filtration, to give necessary conditions for an operator $M$ to be in the centralizer of $L$. Whenever the centralizer equals the algebra generated by $L$ and $M$, we call $L$, $M$ a Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for operators of order $4$ in the first Weyl algebra. Moreover, for true rank $r$ pairs, by means of differential subresultants, we effectively compute the fiber of the rank $r$ spectral sheaf over their spectral curve.

Integrable hierarchies in the [formula omitted]-matrices related to powers of the shift operator

Inside the algebra LTN(R) of N×N-matrices with coefficients from a commutative algebra R over k=R or ℂ, that possess only a finite number of nonzero diagonals above the central diagonal, we consider two deformations of commutative Lie subalgebras generated by the nth power Sn,n⩾1, of the matrix S of the shift operator and a maximal commutative subalgebra h of gln(k), where the evolution equations of the deformed generators are determined by a set of Lax equations, each corresponding to a different decomposition of LTN(R). This yields the h[Sn]-hierarchy and its strict version. We show that both sets of Lax equations are equivalent to a set of zero curvature equations. Next we introduce two Cauchy problems linked with these sets of zero curvature equations and present sufficient conditions under which they can be solved. Moreover, we show that these conditions hold in the formal power series context. Next we introduce two LTN(R)-models, one for each hierarchy, a set of equations in each module and special vectors satisfying these equations from which the Lax equations of each hierarchy can be derived. We conclude by presenting a functional analytic context in which these special vectors can be constructed. Thus one obtains solutions of both hierarchies.

The Maximality of Certain Commutative Subalgebras in Yangians

It is proved that any Bethe subalgebra corresponding to a regular semisimple element in a Yangian is a maximal commutative subalgebra and, moreover, the centralizer of its quadratic part. As a consequence, a description of such subalgebras as the traces of the R-matrix over all finite-dimensional representations of the Yangian is obtained.

Gelfand–Tsetlin Degeneration of Shift of Argument Subalgebras in Types B, C and D

The universal enveloping algebra of any semisimple Lie algebra $$\mathfrak {g}$$ contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of $$\mathfrak {g}$$. For $$\mathfrak {g}=\mathfrak {gl}_n$$ the Gelfand–Tsetlin commutative subalgebra in $$U(\mathfrak {g})$$ arises as some limit of subalgebras from this family. We study the analogous limit of shift of argument subalgebras for classical Lie algebras ($$\mathfrak {g}=\mathfrak {sp}_{2n}$$ or $$\mathfrak {so}_{n}$$). The limit subalgebra is described explicitly in terms of Bethe subalgebras in twisted Yangians $$Y^-(2)$$ and $$Y^+(2)$$, respectively. We index the eigenbasis of such limit subalgebra in any irreducible finite-dimensional representation of $$\mathfrak {g}$$ by Gelfand–Tsetlin patterns of the corresponding type, and conjecture that this indexing is, in appropriate sense, natural. According to Halacheva et al. (Crystals and monodromy of Bethe vectors. arXiv:1708.05105, 2017) such eigenbasis has a natural $$\mathfrak {g}$$-crystal structure. We conjecture that this crystal structure coincides with that on Gelfand–Tsetlin patterns defined by Littelmann in Cones, crystals, and patterns (Transform Groups 3(2):145–179, 1998).

Supersymmetric Polynomials and the Center of the Walled Brauer Algebra

We study a commuting family of elements of the walled Brauer algebra $B_{r,s}(\delta)$, called the Jucys-Murphy elements, and show that the supersymmetric polynomials in these elements belong to the center of the walled Brauer algebra. When $B_{r,s}(\delta)$ is semisimple, we show that those supersymmetric polynomials generate the center. Under the same assumption,we define a maximal commutative subalgebra of $B_{r,s}(\delta)$, called the \emph{Gelfand-Zetlin subalgebra}, and show that it is generated by the Jucys-Murphy elements. As an application, we construct a complete set of primitive orthogonal idempotents of $B_{r,s}(\delta)$, when it is semisimple. We also give an alternative proof of a part of the classification theorem of blocks of $B_{r,s}(\delta)$ in non-semisimple cases, which appeared in the work of Cox-De~Visscher-Doty-Martin.Finally, we present an analogue of Jucys-Murpy elements for the quantized walled Brauer algebra $H_{r,s}(q,\rho)$ over $\mathbb C(q, \rho)$ and by taking the classical limit we show that the supersymmetric polynomials in these elements generates the center. It follows that H. Morton conjecture, which appeared in the study of the relation between the framed HOMFLY skein on the annulus and that on the rectangle with designated boundary points, holds if we extend the scalar from $\mathbb Z[q^{\pm1},\rho^{\pm1}]_{(q-q^{-1})}$ to $\mathbb C(q, \rho)$.

Branches on division algebras

We describe the set of maximal orders in a 2-by-2 matrix algebra over a non-commutative local division algebra B containing a given suborder, for certain important families of such suborders, including rings of integers of division subalgebras of B or most maximal semisimple commutative subalgebras.

Maximal commutative subalgebras of a Grassmann algebra

We provide a new approach to the investigation of maximal commutative subalgebras (with respect to inclusion) of Grassmann algebras. We show that finding a maximal commutative subalgebra in Grassmann algebras is equivalent to constructing an intersecting family of subsets of various odd sizes in [Formula: see text] which satisfies certain combinatorial conditions. Then we find new maximal commutative subalgebras in the Grassmann algebra of odd rank [Formula: see text] by constructing such combinatorial systems for odd [Formula: see text]. These constructions provide counterexamples to conjectures made by Domoskos and Zubor.

A Family of Maximal Algebras of Block Toeplitz Matrices

Abstract The maximal commutative subalgebras containing only Toeplitz matrices have been identified as generalized circulants. A similar simple description cannot be obtained for block Toeplitz matrices. We introduce and investigate certain families of maximal commutative algebras of block Toeplitz matrices.

The commutative core of a Leavitt path algebra

For any unital commutative ring R and for any graph E, we identify the commutative core of the Leavitt path algebra of E with coefficients in R, which is a maximal commutative subalgebra of the Leavitt path algebra. Furthermore, we are able to characterize injectivity of representations which gives a generalization of the Cuntz–Krieger uniqueness theorem.

An algebra which is a maximal commutative subalgebra in very few algebras

“Very few” in the title means two. We show that a unital real or complex algebra generated by a nilpotent of order two can be a maximal abelian subalgebra only in two algebras. One of them is of dimension three and the other of dimension four.

A non-commutative Banach algebra whose maximal commutative subalgebras are all mutually isomorphic

A non-commutative Banach algebra whose maximal commutative subalgebras are all mutually isomorphic

The cycline subalgebra of a Kumjian-Pask algebra

Let $\Lambda$ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalgebra $\mathcal{M}$ inside the Kumjian-Pask algebra ${\rm KP}_R(\Lambda)$. We also prove a generalized Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras which says that a representation of ${\rm KP}_R(\Lambda)$ is injective if and only if it is injective on $\mathcal{M}$.

The [formula omitted] Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

The kernel of the Z2n Dirac–Dunkl operator is examined. The symmetry algebra An of the associated Dirac–Dunkl equation on Sn−1 is determined and is seen to correspond to a higher rank generalization of the Bannai–Ito algebra. A basis for the polynomial null-solutions of the Dirac–Dunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of An and are given explicitly in terms of Jacobi polynomials. The symmetry algebra is shown to act irreducibly on this basis via raising/lowering operators. A scalar realization of An is proposed.