Feigin-Frenkel center in types B, C and D

For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine vertex algebras V_{crit}(g) associated with the simple Lie algebras g of types B, C and D. The construction relies on the Schur-Weyl duality involving the Brauer algebra, and the generators are expressed as weighted traces over tensor spaces and, equivalently, as traces over the spaces of singular vectors for the action of the Lie algebra sl_2 in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g), and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra g. We also introduce analogues of the Bethe subalgebras of the Yangians Y(g) and show that their graded images coincide with the respective commutative subalgebras of U(g[t]).Comment: 29 pages, constructions of Pfaffian-type Sugawara operators and commutative subalgebras in universal enveloping algebras are adde

Homological smoothness and deformations of generalized Weyl algebras

It is an immediate conclusion from Bavula's papers \cite{Bavula:GWA-def}, \cite{Bavula:GWA-tensor-product} that if a generalized Weyl algebra A=\kk[z;\lambda,\eta,\varphi(z)] is homologically smooth, then the polynomial $\varphi(z)$ has no multiple roots. We prove in this paper that the converse is also true. Moreover, formal deformations of $A$ are studied when \kk is of characteristic zero.Comment: Final version, 36 page

Paving over arbitrary MASAs in von Neumann algebras

We consider a paving property for a maximal abelian *-subalgebra (MASA) $A$ in a von Neumann algebra $M$, that we call so-paving, involving approximation in the so-topology, rather than in norm (as in classical Kadison-Singer paving). If $A$ is the range of a normal conditional expectation, then so-paving is equivalent to norm paving in the ultrapower inclusion $A^\omega\subset M^\omega$. We conjecture that any MASA in any von Neumann algebra satisfies so-paving. We use [MSS13] to check this for all MASAs in $\mathcal B(\ell^2 \mathbb N)$, all Cartan subalgebras in amenable von Neumann algebras and in group measure space II$_1$ factors arising from profinite actions. By [P13], the conjecture also holds true for singular MASAs in II$_1$ factors, and we obtain here an improved paving size $C\varepsilon^{-2}$, which we show to be sharp.Comment: v3: Minor changes, final version, to appear in Analysis and PDE. v2: Minor corrections, plus addition of a complete characterization of MASAs that are norm pavabl

Integrable approach to simple exclusion processes with boundaries. Review and progress

We study the matrix ansatz in the quantum group framework, applying integrable systems techniques to statistical physics models. We start by reviewing the two approaches, and then show how one can use the former to get new insight on the latter. We illustrate our method by solving a model of reaction-diffusion. An eigenvector for the transfer matrix for the XXZ spin chain with non-diagonal boundary is also obtained using a matrix ansatz.Comment: 44 page

An equivariant discrete model for complexified arrangement complements

We define a partial ordering on the set $\mathcal {Q}=\mathcal {Q}(\mathsf {M})$ of pairs of topes of an oriented matroid $\mathsf {M}$, and show the geometric realization $\vert\mathcal {Q}\vert$ of the order complex of $\mathcal {Q}$ has the same homotopy type as the Salvetti complex of $\mathsf {M}$. For any element $e$ of the ground set, the complex $\vert\mathcal {Q}_e\vert$ associated to the rank-one oriented matroid on $\{e\}$ has the homotopy type of the circle. There is a natural free simplicial action of $\mathbb{Z}_4$ on $\vert\mathcal {Q}\vert$, with orbit space isomorphic to the order complex of the poset $\mathcal {Q}(\mathsf {M},e)$ associated to the pointed (or affine) oriented matroid $(\mathsf {M},e)$. If $\mathsf {M}$ is the oriented matroid of an arrangement $\mathscr {A}$ of linear hyperplanes in $\mathbb{R}^n$, the $\mathbb{Z}_4$ action corresponds to the diagonal action of $\mathbb{C}^*$ on the complement $M$ of the complexification of $\mathscr {A}$: $\vert\mathcal {Q}\vert$ is equivariantly homotopy-equivalent to $M$ under the identification of $\mathbb{Z}_4$ with the multiplicative subgroup $\{\pm 1, \pm i\}\subset \mathbb{C}^*$, and $\vert\mathcal {Q}(\mathsf {M},e)\vert$ is homotopy- equivalent to the complement of the decone of $\mathscr {A}$ relative to the hyperplane corresponding to $e$. All constructions and arguments are carried out at the level of the underlying posets.We also show that the class of fundamental groups of such complexes is strictly larger than the class of fundamental groups of complements of complex hyperplane arrangements. Specifically, the group of the non- Pappus arrangement is not isomorphic to any realizable arrangement group. The argument uses new structural results concerning the degree-one resonance varieties of small matroids

Grade filtration of linear functional systems

The grade filtration of a finitely generated left module M over an Auslander regular ring D is a built-in classification of the elements of M in terms of their grades (or their (co)dimensions if D is also a Cohen-Macaulay ring). In this paper, we show how grade filtration can be explicitly characterized by means of elementary methods of homological algebra. Our approach avoids the use of sophisticated methods such as bidualizing complexes, spectral sequences, associated cohomology, and Spencer cohomology used in the literature of algebraic analysis. Efficient implementations dedicated to the computation of grade filtration can then be easily developed in the standard computer algebra systems (see the Maple package PurityFiltration and the GAP4 package AbelianSystems). Moreover, this characterization of grade filtration is shown to induce a new presentation of the left D-module M which is defined by a block-triangular matrix formed by equidimensional diagonal blocks. The linear functional system associated with the left D-module M can then be integrated in cascade by successively solving inhomogeneous linear functional systems defined by equidimensional homogeneous linear systems of increasing dimension. This equivalent linear system generally simplifies the computation of closed-form solutions of the original linear system. In particular, many classes of underdetermined/overdetermined linear systems of partial differential equations can be explicitly integrated by the packages PurityFiltration and AbelianSystems, but not by computer algebra systems such as Maple.La filtration par grade d'un module à gauche M finiment engendré sur un anneau Auslander-régulier D est une classification intrinsèque des éléments de M en fonction de leurs grades (ou de leurs (co)dimensions si D est aussi un anneau de Cohen-Macaulay). Dans ce papier, nous montrons comment la filtration par grade peut être explicitement caractérisée au moyen de techniques élémentaires d'algèbre homologique. Notre approche évite l'utilisation de techniques sophistiquées telles que les complexes bidualisants, les suites spectrales, la cohomologie associée et la cohomologie de Spencer utilisées dans la littérature d'analyse algébrique. Des implantations efficaces dédiées au calcul de la filtration par grade peuvent alors être facilement développées dans les systèmes standards de calcul formel (voir le package PurityFiltration de Maple et le package AbelianSystems de GAP4). De plus, cette caractérisation de la filtration par grade induit une nouvelle présentation du D-module à gauche M qui est définie par une matrice triangulaire par blocs formée de blocs diagonaux équidimensionnels. Le système linéaire fonctionnel associé au D-module à gauche M peut alors être intégré en cascade par la résolution successive de systèmes linéaires fonctionnels inhomogènes définis par des systèmes linéaires homogènes équidimensionnels de dimension croissante. Ce système linéaire équivalent simplifie généralement le calcul des solutions sous formes closes du système linéaire originel. En particulier, de nombreux systèmes linéaires sur-déterminés/sous-déterminés d'équations aux dérivées partielles peuvent être explicitement intégrés au moyen des packages PurityFiltration et AbelianSystems, alors qu'ils ne peuvent l'être par des systèmes de calcul formel tels que Maple