## ScienceDirect Publication: Journal of Algebra

1. Noncommutative fiber products and lattice models
Publication date: 15 August 2018
Source:Journal of Algebra, Volume 508

Author(s): Jonas T. Hartwig

We establish a connection between the representation theory of certain noncommutative singular varieties and two-dimensional lattice models. Specifically, we consider noncommutative biparametric deformations of the fiber product of two Kleinian singularities of type A. Special examples are closely related to the Schrödinger algebra $S ( 1 )$, the affine Lie algebra $A 1 ( 1 )$, and a finite W-algebra associated to $sl 4$. The algebras depend on two scalars and two polynomials that must satisfy the Mazorchuk–Turowska Equation (MTE), which we re-interpret as a quantization of the ice rule (local current conservation) in statistical mechanics. Solutions to the MTE, previously classified by the author and D. Rosso, can accordingly be expressed in terms of multisets of higher spin vertex configurations on a twisted cylinder. We first reduce the problem of describing the category of weight modules to the case of a single configuration $L$. Secondly, we classify all simple weight modules over the corresponding algebras $A ( L )$, in terms of the connected components of the cylinder minus $L$. Lastly, we prove that $A ( L )$ are crystalline graded rings (as defined by Nauwelaerts and Van Oystaeyen), and describe the center of $A ( L )$ explicitly in terms of $L$. Along the way we prove several new results about twisted generalized Weyl algebras and their simple weight modules.

2. McKay correspondence for semisimple Hopf actions on regular graded algebras, I
Publication date: 15 August 2018
Source:Journal of Algebra, Volume 508

Author(s): K. Chan, E. Kirkman, C. Walton, J.J. Zhang

In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin–Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring $A H$ under such an action arises as an analogue of a coordinate ring of a Kleinian singularity.

3. Schurity and separability of quasiregular coherent configurations
Publication date: 15 September 2018
Source:Journal of Algebra, Volume 510

Author(s): Mitsugu Hirasaka, Kijung Kim, Ilia Ponomarenko

A permutation group is said to be quasiregular if each of its transitive constituents is regular, and a quasiregular coherent configuration can be thought as a combinatorial analog of such a group: the transitive constituents are replaced by the homogeneous components. In this paper, we are interested in the question when the configuration is schurian, i.e., formed by the orbitals of a permutation group, or/and separable, i.e., uniquely determined by the intersection numbers. In these terms, an old result of Hanna Neumann is, in a sense, dual to the statement that the quasiregular coherent configurations with cyclic homogeneous components are schurian. In the present paper, we (a) establish the duality in a precise form and (b) generalize the latter result by proving that a quasiregular coherent configuration is schurian and separable if the groups associated with the homogeneous components have distributive lattices of normal subgroups.

4. Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher
Publication date: 15 August 2018
Source:Journal of Algebra, Volume 508

Author(s): Vincent Astier, Thomas Unger

Using the theory of signatures of hermitian forms over algebras with involution, developed by us in earlier work, we introduce a notion of positivity for symmetric elements and prove a noncommutative analogue of Artin's solution to Hilbert's 17th problem, characterizing totally positive elements in terms of weighted sums of hermitian squares. As a consequence we obtain an earlier result of Procesi and Schacher and give a complete answer to their question about representation of elements as sums of hermitian squares.

5. Fusion systems containing pearls
Publication date: 15 September 2018
Source:Journal of Algebra, Volume 510

Author(s): Valentina Grazian

An $F$-essential subgroup is called a pearl if it is either elementary abelian of order $p 2$ or non-abelian of order $p 3$. In this paper we start the investigation of fusion systems containing pearls: we determine a bound for the order of p-groups containing pearls and we classify the saturated fusion systems on p-groups containing pearls and having sectional rank at most 4.