Ryosuke Sato
In this paper, we study Markov dynamics on unitary duals of compact quantum groups. We construct such dynamics from characters of quantum groups. Then we show that the dynamics have generators, and we give an explicit formula of the generators using the representation theory. Moreover, we construct Markov dynamics on the unitary dual of an inductive limit of compact quantum groups.
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We propose a natural quantized character theory for inductive systems of compact quantum groups based on KMS states on AF-algebras following Stratila-Voiculescu's work (Stratila-Voiculescu, 1975) (or (Enomoto-Izumi, 2016)), and give its serious investigation when the system consists of quantum unitary groups $U_q(N)$ with $q$ in $(0,1)$. The key features of this work are: The "quantized trace" of a unitary representation of a compact quantum group can be understood as a quantized character associated with the unitary representation and its normalized one is captured as a KMS state with respect…
This paper is devoted to characterizing the so-called order isomorphisms intertwining the $L^2$-semigroups of two Dirichlet forms. We first show that every unitary order isomorphism intertwining semigroups is the composition of $h$-transformation and quasi-homeomorphism. In addition, under the absolute continuity condition on Dirichlet forms, every (not necessarily unitary) order isomorphism intertwining semigroups is the composition of $h$-transformation, quasi-homeomorphism, and multiplication by a certain step function.
We study the character theory of inductive limits of $q$-deformed classical compact groups. In particular, we clarify the relationship between the representation theory of Drinfeld-Jimbo quantized universal enveloping algebras and our previous work on the quantized characters. We also apply the character theory to construct Markov semigroups on unitary duals of $SO_q(2N+1)$, $Sp_q(N)$, and their inductive limits.
A theorem of Siebert asserts that if a sequence of semigroups of probability measures on a Lie group G is weakly convergent to a semigroup of the same type, then the corresponding generating functionals are convergent in the weak operator topology in every unitary representation of the group.The aim of this note is to give a simple proof of the theorem and propose some improvements. In particular, we completely avoid employing unitary representations by showing simply that under the same hypothesis the generating functionals are convergent pointwise on C^2(G). As a corollary,the above thesis o…
We will introduce the notion of inductive limits of compact quantum groups as $W^*$-bialgebras equipped with some additional structures. We also formulate their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantized characters associated with a given inductive system of compact quantum groups. As a byproduct, we will give an explicit representation-theoretic interpretation to some transformations that play an important role in the analysis of $q$-central probability measures on the paths in the Gelfand-Tsetlin graph.
A rigged Hilbert space characterisation of the unbounded generators of quantum completely positive (CP) stochastic semigroups is given. The general form and the dilation of the stochastic completely dissipative (CD) equation over the algebra L(H) is described, as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators is constructed.
We construct the quadratic analogue of the boson Fock functor. While in the first order case all contractions on the 1--particle space can be second quantized, the semigroup of contractions that admit a quadratic second quantization is much smaller due to the nonlinearity. Within this semigroup we characterize the unitary and the isometric elements.
Consider probabilistic distributions on the space of infinite Hermitian matrices $Herm(\infty)$ invariant with respect to the unitary group $U(\infty)$. We describe the closure of $U(\infty)$ in the space of spreading maps (polymorphisms) of $Herm(\infty)$, this closure is a semigroup isomorphic to the semigroup of all contractive operators.
Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space
本文研究无限维相空间中基于 Weyl 表示的量子随机游走，分析高斯平均平移形成的自伴压缩半群，并证明其收敛到以无限维谐振子 Hamilton 量为生成元的量子振子演化半群。
We study a family of measures originating from the signatures of the irreducible components of representations of the unitary group, as the size of the group goes to infinity. Given a random signature $λ$ of length $N$ with counting measure $\mathbf{m}$, we obtain a random signature $μ$ of length $N-1$ through projection onto a unitary group of lower dimension. The signature $μ$ interlaces with the signature $λ$, and we record the data of $μ,λ$ in a random rectangular Young diagram $w$. We show that under a certain set of conditions on $λ$, both $\mathbf{m}$ and $w$ converge as $N\to\infty$. W…
Strongly continuous semigroups of unital completely positive maps (i.e. quantum Markov semigroups or quantum dynamical semigroups) on compact quantum groups are studied. We show that quantum Markov semigroups on the universal or reduced C${}^*$-algebra of a compact quantum group which are translation invariant (w.r.t. to the coproduct) are in one-to-one correspondence with Lévy processes on its $*$-Hopf algebra. We use the theory of Lévy processes on involutive bialgebras to characterize symmetry properties of the associated quantum Markov semigroup. It turns out that the quantum Markov semigr…
We show that the set of entries generated by any finite set of doubly stochastic matrices is nowhere dense, in contrast to the cases of stochastic matrices or unitary matrices. In other words, there is no finite universal set of doubly stochastic matrices, even with the weakest notion of universality. Our proof is based on a theorem for topological semigroups with the convergent property. A topological semigroup is convergent if every infinite product converges. We show that in a compact and convergent semigroup, under some restrictions, the closure of every finitely generated subsemigroup can…
We prove a Feynman path integral formula for the unitary group $ \exp(-itL_{v,θ})$, $t\geq 0$, associated with a discrete magnetic Schrödinger operator $L_{v,θ}$ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$ |\exp(-itL_{v,θ})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y), $$ which controls the unitary group uniformly in the potentials in terms of a Schrödinger semigroup, where the potential $\mathrm{deg}$ is the weighted degree function of the graph.
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