arXiv subjects

Alexander Sherman

Publications and source records attributed to Alexander Sherman.

At least 19 recordsLinked to original sources

arXiv2026

Absolutely flat algebras in tensor categories

We study the relation between semisimple algebras, artinian simple algebras, and artinian absolutely flat algebras in ind-completions of tensor categories (rigid abelian monoidal categories). Our main results show that the three types of algebras coincide, if we restrict to commutative algebras in certain (all, if we accept the main conjectures in the field) symmetric tensor categories of moderate growth. We also provide examples to show that in general these notions tend to diverge, contrary to the classical case of algebras over a field. In one appendix we give the first, to the best of our knowledge, example of a tensor category with finite-dimensional morphism spaces but objects of infinite length. In a second appendix we establish a generalisation of Deligne's theorem that tannakian categories are neutral over algebraically closed fields.

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arXiv2026

The Harish-Chandra isomorphism for supersymmetric spaces and ghost distributions

We prove the Harish-Chandra isomorphism theorem for supersymmetric spaces, describing the polynomial algebra of eigenvalues of invariant differential operators. The polynomials obtained satisfy novel invariance conditions, which remain somewhat mysterious. We also prove the Harish-Chandra isomorphism for ghost distributions, which satisfy a `square root' of the invariance conditions coming from invariant differential operators. All proofs are algebraic, and rely on a rank-one reduction argument and the Chevalley restriction theorem.

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arXiv2025

Semisimplifying categorical Heisenberg actions and periodic equivalences

We systematically apply semisimplification functors in modular representation theory. Motivated by the Duflo--Serganova functor in Lie superalgebras, we construct various functors of interest. In the setting of finite groups, we refine the cyclic group Brauer construction and categorify the Glauberman correspondence. In the setting of degenerate categorical Heisenberg actions, we obtain a rich collection of functors which commute with the categorical action. Applied to well-known categorifications of the basic representation and Fock space, our functors give explicit realizations of periodic equivalences for polynomial functors and symmetric groups first studied by Henke-Koenig. This allows us to globalize the equivalences of Henke-Koenig by symmetric monoidal functors. We apply these results to deduce branching properties of certain modular representations of $S_n$.

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arXiv2025

A remarkable functor on $G$-modules

We introduce a new functor on categories of modular representations of reductive algebraic groups. Our functor has remarkable properties. For example it is a tensor functor and sends every standard and costandard object in the principal block to a one-dimensional object. We connect our functor to recent work of Gruber and conjecture that our functor is equivalent to hypercohomology under the equivalence of the Finkelberg-Mirkovic conjecture.

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arXiv2025

Homogeneous spaces in tensor categories

Let $\mathscr{C}$ be a symmetric tensor category of moderate growth, and let $\mathcal{H}\leq\mathcal{G}$ be algebraic groups in $\mathscr{C}$. We prove that the homogeneous space $\mathcal{G}/\mathcal{H}$ exists as a scheme and is of finite type when $\mathscr{C}$ is geometrically reductive and maximally nilpotent, conditions that are conjecturally equivalent to incompressibility. A key tool is the introduction of a Frobenius kernel of an group scheme. We further show that while $\mathcal{G}_0/\mathcal{H}_0$ and $(\mathcal{G}/\mathcal{H})_0$ need not be the same, they are close enough, so that $\mathcal{G}/\mathcal{H}$ is quasi-affine/affine/proper if and only if $\mathcal{G}_0/\mathcal{H}_0$ is.

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arXiv2024

Sylow theorems for supergroups

We introduce Sylow subgroups and $0$-groups to the theory of complex algebraic supergroups, which mimic Sylow subgroups and $p$-groups in the theory of finite groups. We prove that Sylow subgroups are always $0$-groups, and show that they are unique up to conjugacy. Further, we give an explicit classification of $0$-groups which will be very useful for future applications. Finally, we prove an analogue of Sylow's third theorem on the number of Sylow subgroups of a supergroup.

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arXiv2024

On the Cartan-Helgason theorem for supersymmetric pairs

Let $(\mathfrak{g},\mathfrak{k})$ be a supersymmetric pair arising from a finite-dimensional, symmetrizable Kac-Moody superalgebra $\mathfrak{g}$. An important branching problem is to determine the finite-dimensional highest-weight $\mathfrak{g}$-modules which admit a $\mathfrak{k}$-coinvariant, and thus appear as functions in a corresponding supersymmetric space $\mathcal{G}/\mathcal{K}$. This is the super-analogue of the Cartan-Helgason theorem. We solve this problem via a rank one reduction and an understanding of reflections in singular roots, which generalize odd reflections in the theory of Kac-Moody superalgebras. An explicit presentation of spherical weights is provided for every pair when $\mathfrak{g}$ is indecomposable.

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arXiv2023

A queer Kac-Moody construction

We introduce a new, Kac--Moody-flavoured construction for Lie superalgebras, which incorporates phenomena of the type Q (queer) Lie superalgebra. This is done by replacing a maximal even torus by the most general possible Cartan subalgebra for Lie superalgebras, which is a maximal quasitoral subalgebra. The theory is remarkably rigid but nevertheless unveils a new natural class of Lie superalgebras, which we call type Q Kac--Moody (QKM) algebras. We classify finite-growth type Q Kac--Moody algebras, and obtain in a novel way the $d=2$, $\mathcal{N}=1,2,3,4$ twisted superconformal algebras, along with three other new, finite growth Lie superalgebras. Our work also gives a new perspective on the distinctiveness of the Lie superalgebra $\mathfrak{q}(n)$.

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arXiv2023

An anti-maser for quantum-limited cooling of a microwave cavity

The maser, a microwave (MW) analog of the laser, is a well-established method for generating and amplifying coherent MW irradiation with ultra-low noise. This is accomplished by creating a state of population inversion between two energy levels separated by MW frequency. Thermodynamically, such a state corresponds to a small but negative temperature. The reverse condition, where only the lower energy level is highly populated, corresponds to a very low positive temperature. In this work, we experimentally demonstrate how to generate such a state in condensed matter at moderate cryogenic temperatures. This state is then used to efficiently remove microwave photons from a cavity, continuously cooling it to the quantum limit, well below its ambient temperature. Such an "anti-maser" device could be extremely beneficial for applications that would normally require cooling to millikelvin temperatures to eliminate any MW photons. For instance, superconducting MW quantum circuits (such as qubits and amplifiers) could, with the use of this device, operate efficiently at liquid helium temperatures.

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arXiv2023

It takes two spectral sequences

We study the representation theory of the Lie superalgebra $\mathfrak{gl}(1|1)$, constructing two spectral sequences which eventually annihilate precisely the superdimension zero indecomposable modules in the finite-dimensional category. The pages of these spectral sequences, along with their limits, define symmetric monoidal functors on $\mathrm{Rep} (\mathfrak{gl}(1|1))$. These two spectral sequences are related by contragredient duality, and from their limits we construct explicit semisimplification functors, which we explicitly prove are isomorphic up to a twist. We use these tools to prove branching results for the restriction of simple modules over Kac-Moody and queer Lie superalgebras to $\mathfrak{gl}(1|1)$-subalgebras.

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arXiv2022

Localization theorem for homological vector fields

We present a general theorem which computes the cohomology of a homological vector field on global sections of vector bundles over smooth affine supervarieties. The hypotheses and results have the clear flavor of a localization theorem.

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arXiv2022

Ghost distributions on supersymmetric spaces II: basic classical superalgebras

We study ghost distributions on supersymmetric spaces for the case of basic classical Lie superalgebras. We introduce the notion of interlaced pairs, which are those for which both $(\mathfrak{g},\mathfrak{k})$ and $(\mathfrak{g},\mathfrak{k}')$ admit Iwasawa decompositions. For such pairs we define a ghost algebra, generalizing the subalgebra of $\mathcal{U}\mathfrak{g}$ defined by Gorelik. We realize this algebra as an algebra of $G$-equivariant operators on the supersymmetric space itself, and for certain pairs, the `special' ones, we realize our operators as twisted-equivariant differential operators on $G/K$. We additionally show that the Harish-Chandra morphism is injective, compute its image for all rank one pairs, and provide a conjecture for the image when $(\mathfrak{g},\mathfrak{k})$ is interlaced.

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arXiv2022

Splitting quasireductive supergroups and volumes of supergrassmannians

We introduce the notion of splitting subgroups of quasireducitve supergroups, and explain their significance. For $GL(m|n)$, $Q(n)$, and defect one basic classical supergroups, we give explicit splitting subgroups. We further prove they are minimal up to conjugacy, except in the $GL(m|n)$ case where it remains a conjecture. A key tool in the proof is the computation of the volumes of complex supergrassmannians, which is of interest in its own right.

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arXiv2022

On the Grothendieck ring of a quasireductive Lie superalgebra

Given a Lie superalgebra $\mathfrak{g}$ and a maximal quasitoral subalgebra $\mathfrak{h}$, we consider properties of restrictions of $\mathfrak{g}$-modules to $\mathfrak{h}$. This is a natural generalization of the study of characters in the case when $\mathfrak{h}$ is an even maximal torus. We study the case of $\mathfrak{g}=\mathfrak{q}_n$ with $\mathfrak{h}$ a Cartan subalgebra, and prove several special properties of the restriction in this case, including an explicit realization of the $\mathfrak{h}$-supercharacter ring.

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arXiv2022

On the Duflo-Serganova functor for the queer Lie superalgebra

We study the Duflo-Serganova functor $\operatorname{DS}_x$ for the queer Lie superalgebra $\mathfrak{q}_n$ and for all odd $x$ with $[x,x]$ semisimple. For the case when the rank of $x$ is $1$ we give a formula for multiplicities in terms of the arc diagram attached to $\lambda$. Further, we prove that $\operatorname{DS}_x(L)$ is semisimple if $L$ is a simple finite-dimensional module and $x$ is of rank $1$ satisfying $x^2=0$.

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arXiv2022

Ghost distributions on supersymmetric spaces I: Koszul induced superspaces, branching, and the full ghost centre

Given a Lie superalgebra $\mathfrak{g}$, Gorelik defined the anticentre $\mathcal{A}$ of its enveloping algebra, which consists of certain elements that square to the center. We seek to generalize and enrich the anticentre to the context of supersymmetric pairs $(\mathfrak{g},\mathfrak{k})$, or more generally supersymmetric spaces $G/K$. We define certain invariant distributions on $G/K$, which we call ghost distributions, and which in some sense are induced from invariant distributions on $G_0/K_0$. Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from $G$ to a symmetric subgroup $K'$ which is related (and sometimes conjugate) to $K$. We discuss the case of $G\times G/G$ for an arbitrary quasireductive supergroup $G$, where our results prove the existence of a polynomial which determines projectivity of irreducible $G$-modules. Finally, a generalization of Gorelik's ghost centre is defined called the full ghost centre, $\mathcal{Z}_{full}$. For type I basic Lie superalgebras $\mathfrak{g}$ we fully describe $\mathcal{Z}_{full}$, and prove that if $\mathfrak{g}$ contains an internal grading operator, $\mathcal{Z}_{full}$ consists exactly of those elements in $\mathcal{U}\mathfrak{g}$ acting by $\mathbb{Z}$-graded constants on every finite-dimensional irreducible representation.

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arXiv2022

The Duflo-Serganova functor, vingt ans apr\`es

We review old and new results concerning the $DS$ functor and associated varieties for Lie superalgebras. These notions were introduced in the unpublished manuscript arXiv:math/0507198 by Michel Duflo and the third author. This paper includes the results and proofs of the original manuscript, as well as a survey of more recent results.

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arXiv2022

On symmetries of the Duflo-Serganova functor

We discuss several points regarding symmetries of the Duflo-Serganova functor. In particular we give new constructions of Lie superalgebras, Lie supergroups, and associative superalgebras which act on the Duflo-Serganova functor. We connect our work to a computation of Heidersdorf and Weissauer which computed $DS_x$ for a maximal rank $x$ on Kac-modules for $GL(n|n)$, and extend the ideas and results to $P(n)$.

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