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Michael Müger

Publications and source records attributed to Michael Müger.

5 recordsLinked to original sources
arXiv2025

An integral formula for Lie groups, and the Mathieu conjecture reduced to Abelian non-Lie conjectures

We present an explicit integration formula for the Haar integral on a compact connected Lie group. This formula relies on a known decomposition of a compact connected simple Lie group into symplectic leaves, when one views the group as a Poisson Lie group. In this setting the Haar integral is constructed using the Kostant harmonic volume form on the corresponding flag manifold, and explicit coordinates are known for these invariant differential forms. The formula obtained is related to one found by Reshetikhin-Yakimov. Using our integration formula, we reduce the Mathieu conjecture to two stronger conjectures about Laurent polynomials in several complex variables with polynomial coefficients in several real variable polynomials. In these stronger conjectures there is no reference to Lie group theory.

math.GR↗
arXiv2019

On the moments of a polynomial in one variable

Let $f$ be a non-zero polynomial with complex coefficients and define $M_n(f)=\int_0^1f(x)^n\,dx$. We use ideas of Duistermaat and van der Kallen to prove $\limsup_{n\rightarrow\infty}|M_n(f)|^{1/n}>0$. In particular, $M_n(f)\ne 0$ for infinitely many $n\in{\mathbb N}$.

math.CV↗
arXiv2016

On Ikehara type Tauberian theorems with $O(x^γ)$ remainders

Motivated by analytic number theory, we explore remainder versions of Ikehara's Tauberian theorem yielding power law remainder terms. More precisely, for $f:[1,\infty)\rightarrow{\mathbb R}$ non-negative and non-decreasing we prove $f(x)-x=O(x^γ)$ with $γ<1$ under certain assumptions on $f$. We state a conjecture concerning the weakest natural assumptions and show that we cannot hope for more.

math.CA↗
arXiv2007

Monoids, Embedding Functors and Quantum Groups

We show that the left regular representation π_l of a discrete quantum group (A,Δ) has the absorbing property and forms a monoid (π_l,\tilde{m},\tildeη) in the representation category Rep(A,Δ). Next we show that an absorbing monoid in an abstract tensor *-category C gives rise to an embedding functor E:C->Vect_C, and we identify conditions on the monoid, satisfied by (π_l,\tilde{m},\tildeη), implying that E is *-preserving. As is well-known, from an embedding functor E: C->\mathrm{Hilb} the generalized Tannaka theorem produces a discrete quantum group (A,Δ) such that C is equivalent to Rep_f(A,Δ). Thus, for a C^*-tensor category C with conjugates and irreducible unit the following are equivalent: (1) C is equivalent to the representation category of a discrete quantum group (A,Δ), (2) C admits an absorbing monoid, (3) there exists a *-preserving embedding functor E: C->\mathrm{Hilb}.

math.QA↗