Alexander Alldridge

Alexander Alldridge
University of California, Berkeley | UCB · Department of Mathematics

PD Dr. rer. nat.

About

28
Publications
2,808
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169
Citations
Introduction
I am a mathematician on a Heisenberg Fellowship granted by the German Research Council (DFG). My current research focuses on one hand on the interplay of Harmonic Analysis, Geometry, and Representation Theory, especially for symmetric superspaces and Lie supergroups; on the other hand, I am working on a mathematical theory of disordered topological insulators and superconductors using Noncommutative Geometry.
Additional affiliations
October 2016 - present
University of Cologne
Position
  • Heisenberg Fellowship
September 2010 - September 2016
University of Cologne
Position
  • Group Leader
April 2004 - September 2010
Universität Paderborn
Position
  • Research Assistant
Education
September 1999 - October 2004
Philipps University of Marburg
Field of study
  • Mathematics

Publications

Publications (28)
Article
Full-text available
Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C\...
Preprint
Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$...
Chapter
Let l:= q(n) × q(n), where q(n) denotes the queer Lie superalgebra. The associative superalgebra V of type Q(n) has a left and right action of q(n), and hence is equipped with a canonical l-module structure. We consider a distinguished basis {D λ } of the algebra of l-invariant super-polynomial differential operators on V, which is indexed by stric...
Article
Full-text available
Let $\mathfrak l:= \mathfrak q(n)\times\mathfrak q(n)$, where $\mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $\mathfrak q(n)$, and hence is equipped with a canonical $\mathfrak l$-module structure. We consider a distinguished basis $\{D_\lambda\}$ of the algebra of...
Article
Full-text available
For the super-hyperbolic space in any dimension, we introduce the non-Euclidean Helgason--Fourier transform. We prove an inversion formula exhibiting residue contributions at the poles of the Harish-Chandra c-function, signalling discrete parts in the spectrum. The proof is based on a detailed study of the spherical superfunctions, using recursion...
Chapter
Kirillov’s orbit philosophy holds for nilpotent Lie supergroups in a narrow sense, but due to the paucity of unitary representations, it falls short of being an effective tool of harmonic analysis in its present form. In this note, we survey an approach using families of coadjoint orbits which remedies this deficiency, at least in relevant examples...
Article
Full-text available
We study actions of Lie supergroups, in particular, the hitherto elusive notion of `orbits through odd (or more general) points'. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy supergroups and orbits at general points. In this setting, we show that...
Article
Full-text available
We compute the Harish-Chandra $c$-function for a generic class of rank-one purely non-compact Riemannian symmetric superspaces $X=G/K$ in terms of Euler $\Gamma$ functions, proving that it is meromorphic. Compared to the even case, the poles of the $c$-function are shifted into the right half-space. We derive the full asymptotic Harish-Chandra seri...
Article
Full-text available
For any Lie supergroup whose underlying Lie group is reductive, we prove an extension of the Casselman-Wallach globalisation theorem: There is an equivalence between the category of Harish-Chandra modules and the category of SF-representations (smooth Fr\'echet representations of moderate growth) whose module of finite vectors is Harish-Chandra. As...
Article
Full-text available
This is the extended version of a survey prepared for publication in the Springer INdAM series. Superbosonisation, introduced by Littelmann-Sommers-Zirnbauer, is a generalisation of bosonisation, with applications in Random Matrix Theory and Condensed Matter Physics. We link the superbosonisation identity to Representation Theory and Harmonic Analy...
Article
Full-text available
We introduce a wide category of superspaces, called locally finitely generated, which properly includes supermanifolds but enjoys much stronger permanence properties, as are prompted by applications. Namely, it is closed under taking finite fibre products (i.e. is finitely complete) and thickenings by spectra of Weil superalgebras. Nevertheless, in...
Article
The classical Cartan-Helgason theorem characterises finite-dimensional spherical representations of reductive Lie groups in terms of their highest weights. We generalise the theorem to the case of a reductive symmetric supergroup pair $(G,K)$ of even type. Along the way, we compute the Harish-Chandra $c$-function of the symmetric superspace $G/K$....
Article
Full-text available
The classical Cartan-Helgason theorem characterises finite-dimensional spherical representations of reductive Lie groups in terms of their highest weights. We generalise the theorem to the case of a reductive symmetric supergroup pair (G,K) of even type. Along the way, we compute the Harish-Chandra c-function of the symmetric superspace G/K. By way...
Article
Full-text available
The superbosonization identity of Littelmann, Sommers and Zirnbauer is a new tool for use in studying universality of random matrix ensembles via supersymmetry, which is applicable to non-Gaussian invariant distributions. We give a new conceptual interpretation of this formula, linking it to harmonic superanalysis of Lie supergroups and symmetric s...
Article
Full-text available
With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre bundle sections) exist. In particular, loop supergroups can be realised globally in this framework. It also provid...
Article
We investigate the Berezin integral of non-compactly supported quantities. In the framework of supermanifolds with corners, we give a general, explicit and coordinate-free repesentation of the boundary terms introduced by an arbitrary change of variables. As a corollary, a general Stokes's theorem is derived - here, the boundary integral contains t...
Article
Full-text available
We define a Fourier transform and a convolution product for functions and distributions on Heisenberg--Clifford Lie supergroups. The Fourier transform exchanges the convolution and a pointwise product, and is an intertwining operator for the left regular representation. We generalize various classical theorems, including the Paley--Wiener--Schwartz...
Article
Full-text available
We study the C$^*$-algebra of Wiener-Hopf operators $A_\Omega$ on a cone $\Omega$ with polyhedral base $P$. As is known, a sequence of symbol maps may be defined, and their kernels give a filtration by ideals of $A_\Omega$, with liminary subquotients. One may define $K$-group valued 'index maps' between the subquotients. These form the $E^1$ term o...
Article
Full-text available
We consider symmetric pairs of Lie superalgebras and introduce a graded Harish-Chandra homomorphism. Generalising results of Harish-Chandra and V. Kac, M. Gorelik, we prove that, assuming reductivity, its image is a certain explicit filtered subalgebra J(\( \mathfrak{a} \)) of the Weyl invariants on a Cartan subspace whose associated graded gr J(\(...
Article
Full-text available
Let G be a Lie supergroup and H a closed subsupergroup. We study the unimodularity of the homogeneous supermanifold G/H, i.e. the existence of G-invariant sections of its Berezinian line bundle. To that end, we express this line bundle as a G-equivariant associated bundle of the principal H-bundle G over G/H. We also study the fibre integration of...
Article
Full-text available
In his recent investigation of a super Teichm\"uller space, Sachse (2007), based on work of Molotkov (1984), has proposed a theory of Banach supermanifolds using the `functor of points' approach of Bernstein and Schwarz. We prove that the the category of Berezin-Kostant-Leites supermanifolds is equivalent to the category of finite-dimensional Molot...
Article
Full-text available
Let D=G/K be an irreducible Hermitian symmetric domain. Then G is contained in a complexification, and there exists a closed complex subsemigroup, the so-called minimal Olshanskii semigroup, of the complexification characterised by the fact that all holomorphic discrete series representations of G extend holomorphically to it. Parallel to the class...
Article
Full-text available
We consider Wiener-Hopf operators with continuous symbol, d efined on the L2 space of a convex cone (w.r.t. Lebesgue measure), and the C· -algebra of bounded operators generated by them. In the (classical) case of a Wiener-Hopf operator on the half line, the property of being Fredholm can b e characterised in terms of the symbol, and if Fremdholmne...
Article
Let (g,k) be a reductive symmetric superpair of even type, i.e. so that there exists an even Cartan subspace a in p. The restriction map S(p^*)^k->S(a^*)^W where W=W(g_0:a) is the Weyl group, is injective. We determine its image explicitly. In particular, our theorem applies to the case of a symmetric superpair of group type, i.e. (k+k,k) with the...
Article
We study the multivariate generalisation of the classical Wiener–Hopf algebra, which is the C∗-algebra generated by the Wiener–Hopf operators, given by convolutions restricted to convex cones. By the work of Muhly and Renault, this C∗-algebra is known to be isomorphic to the reduced C∗-algebra of a certain restricted action groupoid. It admits a co...
Article
We study multivariate generalisations of the classical Wiener–Hopf algebra, which is the C∗-algebra generated by the Wiener–Hopf operators, given by convolutions restricted to convex cones. By the work of Muhly and Renault, this C∗-algebra is known to be isomorphic to the reduced C∗-algebra of a certain restricted action groupoid, given by the acti...
Article
Full-text available
Let $G/K$ be a Hermitian symmetric space of non-compact type. We consider for the so-called minimal Olshanskii semigroup $Gammasubset G^C$, the C$^*$-algebra $T$ generated by all Toeplitz operators $T_f$ on the Hardy space $H^2(Gamma)subset L^2(G)$. We describe the construction of ideals of $T$ associated to boundary strata of the domain $Gamma$. S...

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