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states of the infinite qdeformed Heisenberg ferromagnet”, archived incondmat@babbage.sissa.it #9501123
"... 1,2 Abstract. We set up a general structure for the analysis of “frustrationfree ground states”, or “zeroenergy states”, i.e., states minimizing each term in a lattice interaction individually. The nesting of the finite volume ground state spaces is described by a generalized inductive limit of ob ..."
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Cited by 37 (2 self)
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1,2 Abstract. We set up a general structure for the analysis of “frustrationfree ground states”, or “zeroenergy states”, i.e., states minimizing each term in a lattice interaction individually. The nesting of the finite volume ground state spaces is described by a generalized inductive limit of observable algebras. The limit space of this inductive system has a state space which is canonically isomorphic (as a compact convex set) to the set of zeroenergy states. We show that for Heisenberg ferromagnets, and for generalized valence bond solid states, the limit space is an abelian C*algebra, and all zeroenergy states are translationally invariant or periodic. For the qdeformed spin1/2 Heisenberg ferromagnet in one dimension (i.e., the XXZchain with SqU(2)invariant boundary conditions) the limit space is an extension of the noncommutative algebra of compact operators by two points, corresponding to the “all spins up ” and the “all spins down ” states, respectively. These are the only translationally invariant zeroenergy states. The remaining ones are parametrized by the density matrices on a Hilbert space, and converge weakly to the “all up ” (resp. “all down”) state for shifts to − ∞ (resp.
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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Algebraic quantum hypergroups
, 2006
"... An algebraic quantum group is a multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require ..."
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Cited by 8 (1 self)
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An algebraic quantum group is a multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We construct the dual, just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum hypergroup. We define algebraic quantum hypergroups of compact type and discrete type and we show that these types are dual to each other. The algebraic quantum hypergroups of compact type are essentially the algebraic ingredients of the compact quantum hypergroups as introduced and studied (in an operator algebraic context) by Chapovsky and Vainerman. We will give some basic examples in order to illustrate different aspects of the theory. In a separate note, we will consider more special cases and more complicated examples. In particular, in that note, we will give a general construction procedure and show how known examples of these algebraic quantum hypergroups fit into this framework.
Representations of algebraic quantum groups and reconstruction theorems for tensor categories
"... We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the TannakaKrein reconstruction problem. We show that every concrete semisimple tensor ∗category with conjugates is equivalent to the category of finite dimensional nondegene ..."
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Cited by 8 (4 self)
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We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the TannakaKrein reconstruction problem. We show that every concrete semisimple tensor ∗category with conjugates is equivalent to the category of finite dimensional nondegenerate ∗representations of a discrete algebraic quantum group. Working in the selfdual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and Rmatrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical TannakaKrein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well known general result concerning discrete multiplier Hopf ∗algebras. 1
The Martin boundary of a discrete quantum group
 J. reine und angewandte Math
"... We consider the Markov operator Pφ on a discrete quantum group given by convolution with a qtracial state φ. In the study of harmonic elements x, Pφ(x) = x, we define the Martin boundary Aφ. It is a separable C ∗algebra carrying canonical actions of the quantum group and its dual. We establish a ..."
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Cited by 6 (2 self)
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We consider the Markov operator Pφ on a discrete quantum group given by convolution with a qtracial state φ. In the study of harmonic elements x, Pφ(x) = x, we define the Martin boundary Aφ. It is a separable C ∗algebra carrying canonical actions of the quantum group and its dual. We establish a representation theorem to the effect that positive harmonic elements correspond to positive linear functionals on Aφ. The C ∗algebra Aφ has a natural time evolution, and the unit can always be represented by a KMS state. Any such state gives rise to a u.c.p. map from the von Neumann closure of Aφ in its GNS representation to the von Neumann algebra of bounded harmonic elements, which is an analogue of the Poisson integral. Under additional assumptions this map is an isomorphism which respects the actions of the quantum group and its dual. Next we apply these results to identify the Martin boundary of the dual of SUq(2) with the quantum homogeneous sphere of Podle´s. This result extends and unifies previous results by Ph. Biane and M. Izumi.
Embedding ergodic actions of compact quantum groups on C∗–algebras into quotient spaces
 INT. J. MATH
, 2006
"... The notion of compact quantum subgroup is revisited and an alternative definition is given. Induced representations are considered and a Frobenius reciprocity theorem is obtained. A relationship between ergodic actions of compact quantum groups on C∗–algebras and topological transitivity is investig ..."
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Cited by 5 (3 self)
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The notion of compact quantum subgroup is revisited and an alternative definition is given. Induced representations are considered and a Frobenius reciprocity theorem is obtained. A relationship between ergodic actions of compact quantum groups on C∗–algebras and topological transitivity is investigated. A sufficient condition for embedding such actions in quantum quotient spaces is obtained.
Quantum SO(3) groups quantum group actions on M2. arXiv:0810.0398v1 [math.OA
"... Abstract. Answering a question of Shuzhou Wang we give a description of quantum SO(3) groups of Podle´s as universal objects. We use this result to give a complete classification of all continuous compact quantum group actions on M2. 1. ..."
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Cited by 5 (1 self)
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Abstract. Answering a question of Shuzhou Wang we give a description of quantum SO(3) groups of Podle´s as universal objects. We use this result to give a complete classification of all continuous compact quantum group actions on M2. 1.
Quantum spin chains with quantum group symmetry, preprint  KULTF94/8
"... We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasilocal) tails of the action, we find that there are no actions of a pro ..."
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Cited by 2 (0 self)
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We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasilocal) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice translations. The nonlocality arises from the ordering of factors in the quantum group C*algebra, and can be made onesided, thus allowing semilocal actions on a half chain. Under such actions, localized quantum group invariant elements remain localized. Hence the notion of interactions invariant under the quantum group and also under translations, recently studied by many authors, makes sense even though there is no global action of the quantum group. We consider a class of such quantum group invariant interactions with the property that there is a unique translation invariant ground state. Under weak locality assumptions, its GNS representation carries no unitary representation of the quantum group.
Regular Objects, Multiplicative Unitaries and
, 2002
"... The notion of left (resp. right) regular object of a tensor C ∗ –category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (resp. right) regular object, it can be interpreted as a category of corepresentations (resp. representat ..."
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The notion of left (resp. right) regular object of a tensor C ∗ –category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (resp. right) regular object, it can be interpreted as a category of corepresentations (resp. representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry. Conjugation is discussed in the context of multiplicative unitaries and their associated Hopf C ∗ –algebras. It is shown that the conjugate of a left