arXiv subjects

Erich C. Jauch

Publications and source records attributed to Erich C. Jauch.

4 recordsLinked to original sources

arXiv2026

Galois Rings: Ring-Theoretic Properties and Applications to Coulomb Branches and Affine Hecke Algebras

Galois rings and Galois orders, introduced by Futorny and Ovsienko, are realized as subrings of fixed subrings of skew group (or monoid) rings and have numerous applications in the structure and representation theory of associative algebras. This paper consists of two parts. The first parte investigates ring-theoretic properties that follow from the Galois ring structure alone. In particular, we estabilish natural conditions under which Galois are Ore domains or (semi)prime Goldie rings. We also study several ring-theoretic dimensions and combine the theories of Galois rings and PI-algebras to obtain new structural results. In the second part, we apply these results, together with general techniques from ring theory, to affine Hecke algebras in the sense of Ginzburg, Kapranov, and Vasserot, as well as to spherical Coulomb branch algebras. In particular, we prove that these algebras are Jacobson semiprimitive, satisfy the Nullstellensatz, and determine several of their ring-theoretic dimensions. For affine Hecke algebras, we further prove that they satisfy the maximal Nullstellensatz, are integral over their centers, and determine their T-ideals of polynomial identities, PI-degree, and PI-exponents. For spherical Coulomb branch algebras, we compute the Krull dimension, estabilish that they satisfy the Gelfand-Kirillov conjecture, and prove that they are not PI-algebras

math.RA↗

arXiv2024

Hilbert-Samuel Polynomials for Algebras with Special Filtrations

The notion of multiplicity of a module first arose as consequence of Hilbert's work on commutative algebra, relating the dimension of rings with the degree of certain polynomials. For noncommutative rings, the notion of multiplicity first appeared in the context of modules for the Weyl algebra in Bernstein's solution of the problem of analytic continuation posed by I. Gelfand. The notion was shown to be useful to many more noncommutative rings, especially enveloping algebras, rings of differential operators, and quantum groups. In all these cases, the existence of multiplicity is related to the existence of Hilbert-Samuel polynomials. In this work we give an axiomatic definition of algebras with a notion of multiplicity, which we call very nice and modest algebras. We show, in an abstract setting, how the existence of Hilbert-Samuel polynomials implies the existence of a notion of multiplicity. We apply our results for the category of min-holonomic modules -- a notion which coincides with holonomic modules for simple algebras -- and that shares many similarities with it. In particular, we generalize the usual results in the literature that are stated for Ore domains, in the more general context of prime algebras, and we show that rational Cherednik algebras admit a notion of multiplicity.

math.RA↗

arXiv2024

Maps between standard and principal flag orders

Galois orders, introduced in 2010 by V. Futorny and S. Ovsienko, form a class of associative algebras that contain many important examples, such as the enveloping algebra of $\mathfrak{gl}_n$ (as well as its quantum deformation), generalized Weyl algebras, and shifted Yangians. The main motivation for introducing Galois orders is they provide a setting for studying certain infinite dimensional irreducible representations, called Gelfand-Tsetlin modules. Principal Galois orders, defined by J. Hartwig in 2017, are Galois orders with an extra property, which makes them easier to study. All of the mentioned examples are principal Galois orders. In 2019, B. Webster defined principal flag orders which in most situations are Morita equivalent to principal Galois orders, and further simplifies their study. This paper describes some techniques to connect the study pairs of standard flag and Galois orders with related data. Such techniques include: a sufficient condition for morphisms between standard flag and Galois orders to exist, a property of flag orders related to differential operators on affine varieties, and constructing tensor products of standard and principal flag and Galois orders.

math.RT↗

arXiv2020

An extension of $U(\mathfrak{gl}_n)$ related to the alternating group and Galois orders

In 2010, V. Futorny and S. Ovsienko gave a realization of $U(\mathfrak{gl}_n)$ as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of $S_1\times S_2\times\cdots\times S_n$, where $S_j$ is the symmetric group on $j$ variables. An interesting question is what a similar algebra would be in the invariant ring with respect to a product of alternating groups. In this paper we define such an algebra, denoted $\mathscr{A}(\mathfrak{gl}_n)$, and show that it is a Galois ring. For $n=2$, we show that it is a generalized Weyl algebra, and for $n=3$ provide generators and a list of verified relations. We also discuss some techniques to construct Galois orders from Galois rings. Additionally, we study categories of finite-dimensional modules and generic Gelfand-Tsetlin modules over $\mathscr{A}(\mathfrak{gl}_n)$. Finally, we discuss connections between the Gelfand-Kirillov Conjecture, $\mathscr{A}(\mathfrak{gl}_n)$, and the positive solution to Noether's problem for the alternating group.

math.RT↗