arXiv subjects

Jonas T. Hartwig

Publications and source records attributed to Jonas T. Hartwig.

At least 19 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

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arXiv2026

ABRR Summation Formulas for Relative Extremal Projectors

In 2004, Khoroshkin proved that the extremal projector is equivalent to the dynamical twist. The Arnaudon-Buffenoir-Ragoucy-Roche (ABRR) equation, satisfied by the dynamical twist, yields a recursive formula for the coefficients in the extremal projector. The resulting summation formula is uniform and powerful in its applications to representation theory and Mickelsson-Zhelobenko reduction algebras. In this paper we generalize this recursive formula to the relative extremal projector, introduced by Conley and Sepanski in 2003. When the Levi subalgebra is the Cartan subalgebra, we recover the usual ABRR recursion. We also find a compact and explicit expression for the solution to the recursion. Our setting includes infinite-dimensional contragredient Lie superalgebras, Kac-Moody algebras, basic classical Lie superalgebras, and finite-dimensional reductive Lie algebras.

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arXiv2025

Parabolic Stabilization and Cutting for Reduction Superalgebras

The diagonal reduction algebra of a reductive Lie algebra $\mathfrak{g}$ is a localization of the Mickelsson algebra associated to the symmetric pair $(\mathfrak{g}\times\mathfrak{g},\, \mathfrak{g})$. In 2010, Khoroshkin and Ogievetsky introduced the methods of stabilization and cutting, which relate the commutation relations in the diagonal reduction algebra of $\mathfrak{gl}_m\oplus\mathfrak{gl}_n$ with those in the diagonal reduction algebra of $\mathfrak{gl}_{m+n}$. We extend this method to a wide range of reduction algebras, including all diagonal and differential reduction algebras for basic classical Lie superalgebras. We show how the method can be used for computing relations in the diagonal reduction algebra of $\mathfrak{so}_8$ and differential reduction algebra of $\mathfrak{sp}_{2n}$.

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arXiv2025

Dirac reduction algebra

There is a homomorphism of associative superalgebras from the enveloping algebra of the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ to the Weyl-Clifford superalgebra $W(2n|n)$ with $2n$ even Weyl algebra generators and $n$ odd Clifford algebra generators. Under this homomorphism, the positive odd root vector $x\in\mathfrak{osp}(1|2)$ is sent to the Dirac operator $γ^μ\partial_μ\in W(2n|n)$ and generates a left ideal $I$. The corresponding reduction (super)algebra, denoted $Z_n$, is the normalizer of $I$ in $W(2n|n)$ modulo $I$. By construction, $Z_n$ acts on the space of all Clifford algebra-valued polynomial solutions to the (massless) Dirac equation. In this paper, we find a complete presentation of (a localization of) this so-termed Dirac reduction algebra. Furthermore, we use the Dirac reduction algebra to generate all polynomial solutions to the Dirac equation in $n$-dimensional flat spacetime.

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arXiv2025

Subcategories of Module Categories via Restricted Yoneda Embeddings

We propose a framework for producing interesting subcategories of the category ${}_A\mathsf{Mod}$ of left $A$-modules, where $A$ is an associative algebra over a field $k$. The construction is based on the composition, $Y$, of the Yoneda embedding of ${}_A\mathsf{Mod}$ with a restriction to certain subcategories $\mathcal{B}\subset {}_A\mathsf{Mod}$, typically consisting of cyclic modules. We describe the subcategories on which $Y$ provides an equivalence of categories. This also provides a way to understand the subcategories of ${}_A\mathsf{Mod}$ that arise this way. Many well-known categories are obtained in this way, including categories of weight modules and Harish-Chandra modules with respect to a subalgebra $Γ$ of $A$. In other special cases the equivalence involves modules over the Mickelsson step algebra associated to a reductive pair of Lie algebras.

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arXiv2025

Exact Solutions to the Klein--Gordon Equation via Reduction Algebras

Reduction algebras (also known as generalized Mickelsson algebras, Zhelobenko algebras, or transvector algebras) are well-studied associative algebras appearing in the representation theory of Lie algebras. In the 1990s, Zhelobenko noted that reduction algebras have a connection to field equations from physics, whereas Howe's study of dual pairs in the 1980s signifes that the link originated even earlier. In this paper, we revisit the simplest case of a scalar field, working in arbitrary spacetime dimension $n\ge 3$, and in arbitrary flat metric $η_{ab}$. We recall that the field equations specialize to the homogeneous Laplace equation and the (massless) Klein--Gordon equation for appropriate metrics; correspondingly, there is a representation of the Lie algebra $\mathfrak{sl}_2$ (technically, $\mathfrak{sp}_2$) by differential operators and an associated reduction algebra. The reduction algebra contains raising operators, which provide a means to construct bosonic states $|a_1a_2\cdots a_\ell\rangle$ as certain degree $\ell$ polynomials solving the relevant field equation. We give an explicit formula for these solutions and prove that they span the polynomial part of the solution space. We also give a complete presentation of the reduction algebra and compute the inner product between the bosonic states in terms of the rational dynamical R-matrix.

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arXiv2025

Symplectic Differential Reduction Algebras and Generalized Weyl Algebras

Given a map $Ξ\colon U(\mathfrak{g})\rightarrow A$ of associative algebras, with $U(\mathfrak{g})$ the universal enveloping algebra of a (complex) finite-dimensional reductive Lie algebra $\mathfrak{g}$, the restriction functor from $A$-modules to $U(\mathfrak{g})$-modules is intimately tied to the representation theory of an $A$-subquotient known as the reduction algebra with respect to $(A,\mathfrak{g},Ξ)$. Herlemont and Ogievetsky described differential reduction algebras for the general linear Lie algebra $\mathfrak{gl}(n)$ as algebras of deformed differential operators. Their map $Ξ$ is a realization of $\mathfrak{gl}(n)$ in the $N$-fold tensor product of the $n$-th Weyl algebra tensored with $U(\mathfrak{gl}(n))$. In this paper, we further the study of differential reduction algebras by finding a presentation in the case when $\mathfrak{g}$ is the symplectic Lie algebra of rank two and $Ξ$ is a canonical realization of $\mathfrak{g}$ inside the second Weyl algebra tensor the universal enveloping algebra of $\mathfrak{g}$, suitably localized. Furthermore, we prove that this differential reduction algebra is a generalized Weyl algebra (GWA), in the sense of Bavula, of a new type we term skew-affine. It is believed that symplectic differential reduction algebras are all skew-affine GWAs; then their irreducible weight modules could be obtained from standard GWA techniques.

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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.

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arXiv2024

Galois order realization of noncommutative type $D$ Kleinian singularities

Galois orders, introduced by Futorny and Ovsienko, is a class of noncommutative algebras that includes generalized Weyl algebras, the enveloping algebra of the general linear Lie algebra and many others. We prove that the noncommutative Kleinian singularities of type $D$ can be realized as principal Galois orders. Our starting point is an embedding theorem due to Boddington. We also compute explicit generators for the corresponding (Morita equivalent) flag order, as a subalgebra of the nil-Hecke algebra of type $A_1^{(1)}$. Lastly, we compute structure constants for Harish-Chandra modules of local distributions and give a visual description of their structure from which subquotients are easily obtained.

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arXiv2023

Grothendieck rings of towers of generalized Weyl algebras in the finite orbit case

Previously we showed that the tensor product of a weight module over a generalized Weyl algebra (GWA) with a weight module over another GWA is a weight module over a third GWA. In this paper we compute tensor products of simple and indecomposable weight modules over generalized Weyl algebras supported on a finite orbit. This allows us to give a complete presentation by generators and relations of the Grothendieck ring of the categories of weight modules over a tower of generalized Weyl algebras in this setting. We also obtain partial results about the split Grothendieck ring. We described the case of infinite orbits in previous work.

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arXiv2022

Ghost center and representations of the diagonal reduction algebra of $\mathfrak{osp}(1|2)$

Reduction algebras are known by many names in the literature, including step algebras, Mickelsson algebras, Zhelobenko algebras, and transvector algebras, to name a few. These algebras, realized by raising and lowering operators, allow for the calculation of Clebsch-Gordan coefficients, branching rules, and intertwining operators; and have connections to extremal equations and dynamical R-matrices in integrable face models. In this paper we continue the study of the diagonal reduction superalgebra $A$ of the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$. We construct a Harish-Chandra homomorphism, Verma modules, and study the Shapovalov form on each Verma module. Using these results, we prove that the ghost center (center plus anti-center) of $A$ is generated by two central elements and one anti-central element (analogous to the Scasimir due to Leśniewski for $\mathfrak{osp}(1|2)$). As another application, we classify all finite-dimensional irreducible representations of $A$. Lastly, we calculate an infinite-dimensional tensor product decomposition explicitly.

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arXiv2022

Diagonal reduction algebra for $\mathfrak{osp}(1|2)$

The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras $(\mathfrak{G},\mathfrak{g})$ has previously been considered by Khoroshkin and Ogievetsky in the case of the diagonal reduction algebra for $\mathfrak{gl}(n)$. In this paper we consider the diagonal reduction algebra of the pair of Lie superalgebras $\left(\mathfrak{osp}(1|2) \times \mathfrak{osp}(1|2), \mathfrak{osp}(1|2)\right)$ as a double coset space having an associative diamond product and give a complete presentation in terms of generators and relations. We also provide a PBW basis for this reduction algebra along with Casimir-like elements and a subgroup of automorphisms.

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arXiv2021

Harish-Chandra modules over Hopf Galois orders

The theory of Galois orders was introduced by Futorny and Ovsienko. We introduce the notion of $\mathcal{H}$-Galois $Λ$-orders. These are certain noncommutative orders $F$ in a smash product of the fraction field of a noetherian integral domain $Λ$ by a Hopf algebra $\mathcal{H}$ (or, more generally, by a coideal subalgebra of a Hopf algebra). They are generalizations of Webster's principal flag orders. Examples include Cherednik algebras, as well as examples from Hopf Galois theory. We also define spherical Galois orders, which are the corresponding generalizations of principal Galois orders introduced by the author. The main results are (1) for every maximal ideal $\mathfrak{m}$ of $Λ$ of finite codimension, there exists a simple Harish-Chandra $F$-module in the fiber of $\mathfrak{m}$; (2) for every character of $Λ$ we construct a canonical simple Harish-Chandra module as a subquotient of the module of local distributions; (3) if a certain stabilizer coalgebra is finite-dimensional, then the corresponding fiber of simple Harish-Chandra modules is finite; (4) centralizers of symmetrizing idempotents are spherical Galois orders and every spherical Galois order appears that way.

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arXiv2020

Gelfand-Tsetlin Crystals

We give a crystal structure on the set of Gelfand-Tsetlin patterns which parametrize bases for finite-dimensional irreducible representations of the general linear Lie algebra. The crystal data are given in closed form, expressed using tropical polynomial functions of the entries of the patterns. We prove that with this crystal structure, the natural bijection between Gelfand-Tsetlin patterns and semistandard Young tableaux is a crystal isomorphism.

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arXiv2020

Grothendieck rings of towers of twisted generalized Weyl algebras

Twisted generalized Weyl algebras (TGWAs) $A(R,σ,t)$ are defined over a base ring $R$ by parameters $σ$ and $t$, where $σ$ is an $n$-tuple of automorphisms, and $t$ is an $n$-tuple of elements in the center of $R$. We show that, for fixed $R$ and $σ$, there is a natural algebra map $A(R,σ,tt')\to A(R,σ,t)\otimes_R A(R,σ,t')$. This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all $t$) of the Grothendieck groups of the categories of weight modules for $A(R,σ,t)$. We give presentations of these Grothendieck rings for $n=1,2$, when $R=\mathbb{C}[z]$. As a consequence, for $n=1$, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over $\mathfrak{sl}_2$ is a tensor product of two Weyl algebra modules.

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arXiv2019

Dynamical Decomposition of Bilinear Control Systems subject to Symmetries

We describe a method to analyze and decompose the dynamics of a control system on a Lie group subject to symmetries. The method is based on the concept of generalized Young symmetrizers of representation theory. It naturally applies to the situation where the system evolves on a tensor product space and there exists a finite group of symmetries for the dynamics which interchanges the various factors. This is the case for quantum mechanical multipartite systems, such as spin networks, where each factor of the tensor product represents the state of one of the component systems. We present several examples of applications and indicate directions for future research.

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arXiv2019

Classification of twisted generalized Weyl algebras over polynomial rings

Let $R$ be a polynomial ring in $m$ variables over a field of characteristic zero. We classify all rank $n$ twisted generalized Weyl algebras over $R$, up to $\mathbb{Z}^n$-graded isomorphisms, in terms of higher spin 6-vertex configurations. Examples of such algebras include infinite-dimensional primitive quotients of $U(\mathfrak{g})$ where $\mathfrak{g}=\mathfrak{gl}_n$, $\mathfrak{sl}_n$, or $\mathfrak{sp}_{2n}$, algebras related to $U(\widehat{\mathfrak{sl}}_2)$ and a finite W-algebra associated to $\mathfrak{sl}_4$. To accomplish this classification we first show that the problem is equivalent to classifying solutions to the binary and ternary consistency equations. Secondly, we show that the latter problem can be reduced to the case $n=2$, which can be solved using methods from previous work by the authors. As a consequence we obtain the surprising fact that (in the setting of the present paper) the ternary consistency relation follows from the binary consistency relation.

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arXiv2018

Clifford and Weyl superalgebras and spinor representations

We construct a family of twisted generalized Weyl algebras which includes Weyl-Clifford superalgebras and quotients of the enveloping algebras of $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(m|2n)$. We give a condition for when a canonical representation by differential operators is faithful. Lastly, we give a description of the graded support of these algebras in terms of pattern-avoiding vector compositions.

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