Periodicities in Cluster Algebras and Cluster Automorphism Groups

We study the relations between two groups related to cluster automorphism groups which are defined by Assem,Schiffler and Shamchenko.We establish the relation-ships among (strict) direct cluster automorphism groups and those groups consisting of periodicities of labeled seeds and exchange matrices,respectively,in the language of short exact sequences.As an application,we characterize automorphism-finite cluster algebras in the cases of bipartite seeds or finite mutation type.Finally,we study the relation between the group Aut(A) for a cluster algebra A and the group AutMn(S) for a mutation group Mn and a labeled mutation class S,and we give a negative answer via counter-examples to King and Pressland's problem.     

Alternative routes to equivalent classical models of a quantum system

<正>Coarse-graining of some sort is a fundamental and unavoidable step in any attempt to derive the classical mechanical behavior from the quantum formalism.We utilize the two-mode Bose-Hubbard model to illustrate how different coarse-grained systems can be naturally associated with a fixed quantum system if it is compatible with different dynamical algebras.Alternative coarse-grained systems generate different evolutions of the same physical quantities,and the difference becomes negligible only in the appropriate macro-limit.     

Jordan operator algebras revisited

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . In two recent papers by the authors and Neal, a theory for these spaces was developed. It was shown there that much of the theory of associative operator algebras, in particular, surprisingly much of the associative theory from several recent papers of the first author and coauthors, generalizes to Jordan operator algebras. In the present paper we complete this task, giving several results which generalize the associative case in these papers, relating to unitizations, real positivity, hereditary subalgebras, and a couple of other topics. We also solve one of the three open problems stated at the end of our earlier joint paper on Jordan operator algebras.     

On actions of Drinfel'd doubles on finite dimensional algebras

Let q be an nth root of unity for $n>2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-Jürgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an n-dimensional associative algebra A. They further showed that each such module structure extends uniquely to make A a module algebra over the Drinfel'd double of $T_n(q)$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras H “close” to the Taft algebras on finite-dimensional algebras analogous to A above. Such Hopf algebras H include the Sweedler (Hopf) algebra of dimension 4, bosonizations of quantum linear spaces, and the Frobenius–Lusztig kernel $u_q({\mathfrak{sl}}_2)$.