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Actions and coactions of finite quantum groupoids on von Neumann algebras, extensions of the matched pair procedure ( submitted to
 28 J.M. Vallin : C ∗ algèbres de Hopf et C ∗ algèbres de Kac. Pro.London.Math.Soc 50, No.3
, 1985
"... Abstract. In this work we investigate the notion of action or coaction of a finite quantum groupoid in von Neumann algebras context. In particular we prove a double crossed product theorem and prove the existence of an universal von Neumann algebra on which any finite groupoids acts outerly. In prev ..."
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Abstract. In this work we investigate the notion of action or coaction of a finite quantum groupoid in von Neumann algebras context. In particular we prove a double crossed product theorem and prove the existence of an universal von Neumann algebra on which any finite groupoids acts outerly. In previous works, N. Andruskiewitsch and S.Natale define for any match pair of groupoids two C ∗quantum groupoids in duality, we give here an interpretation of them in terms of crossed products of groupoids using a multiplicative partial isometry which gives a complete description of these structures. In a next work we shall give a third description of these structures dealing with inclusions of depth two inclusions of von Neumann algebras associated with outer actions of match pairs of groupoids, and a study, in the same spirit, of an other extension of the match pair procedure. 1991 Mathematics Subject Classification. 17B37,46L35. Key words and phrases. Multiplicative partial isometries, groupoids, subfactors.
Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
, 2009
"... A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromod ..."
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A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic JahnTeller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non–Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier–Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasitriangular, quasiHopf algebras, bialgebroids, GrassmannHopf algebras and Higher Dimensional Algebra. On the one hand, this quantum
ON THE QUIVERTHEORETICAL QUANTUM YANGBAXTER EQUATION
, 2004
"... Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum YangBaxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution ” for short. Results of Eting ..."
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Cited by 2 (1 self)
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Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum YangBaxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution ” for short. Results of EtingofSchedlerSoloviev, LuYanZhu and Takeuchi on the settheoretical quantum YangBaxter equation are generalized to the context of quivers, with groupoids playing the rôle of groups. The notion of “braided groupoid ” is introduced. Braided groupoids are solutions and are characterized in terms of bijective 1cocycles. The structure groupoid of a nondegenerate solution is defined; it is shown that it is braided groupoid. The reduced structure groupoid of a nondegenerate solution is also defined. Nondegenerate solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct startriangular face models and realize them as modules over quasitriangular quantum groupoids introduced in recent papers by M. Aguiar, S. Natale and the author.
ON BRAIDED GROUPOIDS
, 2005
"... Abstract. We study and give examples of braided groupoids, and a fortiori, nondegenerate solutions of the quivertheoretical braid equation. ..."
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Abstract. We study and give examples of braided groupoids, and a fortiori, nondegenerate solutions of the quivertheoretical braid equation.
QUANTUM SYMMETRIES, OPERATOR ALGEBRA AND QUANTUM GROUPOID REPRESENTATIONS: PARACRYSTALLINE SYSTEMS, TOPOLOGICAL ORDER, SUPERSYMMETRY AND GLOBAL SYMMETRY BREAKING
, 2011
"... Novel approaches to extended quantum symmetry, paracrystals, quasicrystals, noncrystalline solids, topological order, supersymmetry and spontaneous, global symmetry breaking are outlined in terms of quantum groupoid, quantum double groupoids and dual, quantum algebroid structures. Physical applicati ..."
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Novel approaches to extended quantum symmetry, paracrystals, quasicrystals, noncrystalline solids, topological order, supersymmetry and spontaneous, global symmetry breaking are outlined in terms of quantum groupoid, quantum double groupoids and dual, quantum algebroid structures. Physical applications of such quantum groupoid and quantum algebroid representations to quasicrystalline structures and paracrystals, quantum gravity, as well as the applications of the Goldstone and Noether's theorems to: phase transitions in superconductors/superfluids, ferromagnets, antiferromagnets, mictomagnets, quasiparticle (nucleon) ultrahot plasmas, nuclear fusion, and the integrability of quantum systems are also considered. Both conceptual developments and novel approaches to Quantum theories are here proposed starting from existing Quantum Group Algebra (QGA), Algebraic Quantum Field Theories (AQFT), standard and effective Quantum Field Theories (QFT), as well as the refined `machinery' of
ON THE QUIVERTHEORETICAL QUANTUM YANGBAXTER EQUATION
"... Abstract. Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum YangBaxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution ” for short. Results ..."
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Abstract. Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum YangBaxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution ” for short. Results of EtingofSchedlerSoloviev, LuYanZhu and Takeuchi on the settheoretical quantum YangBaxter equation are generalized to the context of quivers, with groupoids playing the role of groups. The notion of “braided groupoid ” is introduced. Braided groupoids are solutions and are characterized in terms of bijective 1cocycles. The structure groupoid of a nondegenerate solution is defined; it is shown that it is braided groupoid. The reduced structure groupoid of a nondegenerate solution is also defined. Nondegenerate solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct startriangular face models and realize them as modules over quasitriangular quantum groupoids introduced in papers by M. Aguiar, S. Natale and the author.
DEFORMATIONS OF A MATCHED PAIR AND SCHREIER TYPE THEOREMS FOR BICROSSED PRODUCT OF GROUPS
, 903
"... Abstract. We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups (H, G, α, β) is deformed using a combinatorial datum (σ, v, r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: G → ..."
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Abstract. We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups (H, G, α, β) is deformed using a combinatorial datum (σ, v, r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: G → H in order to obtain a new matched pair ` H,(G, ∗), α ′ , β ′ ´ such that there exist an σinvariant isomorphism of groups H α⊲⊳β G ∼ = H α ′⊲⊳β ′ (G, ∗). Moreover, if we fix the group H and the automorphism σ ∈ Aut(H) then any σinvariant isomorphism H α⊲⊳β G ∼ = H α ′ ⊲⊳β ′ G′ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed product of groups are given.