Associated to an Hadamard matrix $H ∈ M_N(ℂ)$ is the spectral measure μ ∈ 𝓟[0,N] of the corresponding Hopf image algebra, A = C(G) with $G ⊂ S⁺_N$. We study a certain family of discrete measures $μ^r ∈ 𝓟[0,N]$, coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type $∫_0^N (x/N)^p dμ^r(x) = ∫_0^N (x/N)^r dν^p(x)$, where $μ^r,ν^r$ are the truncations of the spectral measures μ,ν associated to $H,H^t$. We also prove, using these truncations $μ^r,ν^r$, that for any deformed Fourier matrix $H=F_M ⊗_Q F_N$ we have μ = ν.
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements about matrix models.
3
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if π:A → Mₙ(ℂ) is a finite-dimensional representation of a Hopf C*-algebra, we prove that the idempotent state associated to its Hopf image A' must be the convolution Cesàro limit of the linear functional φ = tr ∘ π. We then discuss some consequences of this result, notably to inner linearity questions.
4
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks and comments.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.