Introduction. Let G be a reductive Lie group with maximal compact subgroup K and let Γ be a cocompact discrete subgroup of G. We consider the induced representation IndΓ (C), which is the canonical representation on the space L2(Γ\G). The duality theorem of Gelfand [6] states that there is a Frobenius reciprocity as follows: for any irreducible unitary representation π of G it holds HomG,cont(I...

While multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols, obtaining a high-level, structural understanding of entanglement involving arbitrarily many qubits is a longstanding open problem in quantum computer science. In this paper we expose the algebraic and graphical structure of the GHZ-state and the W-state, as well as a purely graphical...

In the present paper we give a self-contained proof of Imprimitivity theorem for systems of covariance, or generalised imprimitivity systems, based on transitive spaces. The theorem holds for locally compact groups and non-normalised positive operator valued (POV) measures. For projective valued measures, the theorem was proven by Mackey, [21], for separable groups, and by Blattner, [4], in ful...

We generalize the Perron–Frobenius Theorem for nonnegative matrices to the class of nonnegative tensors.

In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commu...

We introduce complex cones and associated projective gauges, generalizing a real Birkhoff cone and its Hilbert metric to complex vector spaces. We deduce a variety of spectral gap theorems in complex Banach spaces. We prove a dominated complex cone-contraction Theorem and use it to extend the classical Perron-Frobenius Theorem to complex matrices, Jentzsch’s Theorem to complex integral operator...

Starting from the quantum differential equation associated to a weighted projective space, which is given by Coates, Corti, Lee and Tseng, we construct a Frobenius manifold. We see that the Frobenius manifold coincides with the big quantum cohomology of the weighted projective space. The construction is based on Dubrovin’s reconstruction theorem.