In this work,we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=∈_(ji)~kθ_k and a momentum noncommutativity matrix paramet...In this work,we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=∈_(ji)~kθ_k and a momentum noncommutativity matrix parameter β_(ij)=∈_(ij)~kβ_k,is shown to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS).Imposing some constraints on this particular transformation,we firstly find that the product of the two parameters θ and β possesses a lower bound in direct relation with Heisenberg incertitude relations,and secondly that the two parameters are equivalent but with opposite sign,up to a dimension factor depending on the physical system under study.This means that noncommutativity is represented by a unique parameter which may play the role of a fundamental constant characterizing the whole NCQPS.Within our framework,we treat some physical systems on NCQPS:free particle,harmonic oscillator,system of two-charged particles,Hydrogen atom.Among the obtained results, we discover a new phenomenon which consists of a free particle on NCQPS viewed as equivalent to a harmonic oscillator with Larmor frequency depending on β,representing the same particle in presence of a magnetic field=q~(-1).For the other examples,additional correction terms depending on β appear in the expression of the energy spectrum.Finally,in the two-particle system case,we emphasize the fact that for two opposite charges noncornmutativity is effectively feeled with opposite sign.展开更多
In this paper,we give the notion of derivations of Lie 2-algebras using explicit formulas,and construct the associated derivation Lie 3-algebra.We prove that isomorphism classes of non-abelian extensions of Lie 2-alge...In this paper,we give the notion of derivations of Lie 2-algebras using explicit formulas,and construct the associated derivation Lie 3-algebra.We prove that isomorphism classes of non-abelian extensions of Lie 2-algebras are classified by equivalence classes of morphisms from a Lie 2-algebra to a derivation Lie 3-algebra.展开更多
Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aω of rank m there are a completely irration...Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aω of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr（Aω） = tr（Aρ）. It is proved that the set of all C*-algebras of sections of locally trivial C*-algebra bundles over S2 with fibres Aω. has a group structure, denoted by π1（Aut（Aω.））, which is isomorphic to Z if d > 1 and {0} if d > 1. Let Bcd be a cd-homogeneous C*-algebra over S2 x T2 of which no non-trivial matrix algebra can be factored out. The spherical noncommutative torns Sρcd is defined by twisting C*（T2 x Zm-2） in Bcd C* （Z（m-2）） by a totally skew multiplier ρ on T2 x Z（m-2）. It is shown that Sρcd Mp∞ is isomorphic to C（S2） C* （T2 x Zm-2, ρ） Mcd（C） Mp∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.展开更多
This short paper is based on the talk on the conference Operator Algebras and Related Topics held on July 23-27, 2010, Beijing. The author surveys recent developments of the noncommutative gravity in joint works with ...This short paper is based on the talk on the conference Operator Algebras and Related Topics held on July 23-27, 2010, Beijing. The author surveys recent developments of the noncommutative gravity in joint works with Chaichian, Tureanu, Sun, Wang, Xie and Zhang.展开更多
The generalized noncommutative torus Tkp of rank n was defined in [4] by the crossed product Am/k ×a3 Z ×a4 … ×an Z, where the actions ai of Z on the fibre Mk(C) of a rational rotation algebra Am/k are...The generalized noncommutative torus Tkp of rank n was defined in [4] by the crossed product Am/k ×a3 Z ×a4 … ×an Z, where the actions ai of Z on the fibre Mk(C) of a rational rotation algebra Am/k are trivial, and C*(kZ × kZ) ×a3 Z ×a4 ... ×an Z is a completely irrational noncommutative torus Ap of rank n. It is shown in this paper that Tkp is strongly Morita equivalent to Ap, and that Tkp (?) Mp∞ is isomorphic to Ap (?) Mk(C) (?) Mp∞ if and only if the set of prime factors of k is a subset of the set of prime factors of p.展开更多
In this expository paper,we describe the study of certain non-self-adjoint operator algebras,the Hardy algebras,and their representation theory.We view these algebras as algebras of (operator valued) functions on thei...In this expository paper,we describe the study of certain non-self-adjoint operator algebras,the Hardy algebras,and their representation theory.We view these algebras as algebras of (operator valued) functions on their spaces of representations.We will show that these spaces of representations can be parameterized as unit balls of certain W*-correspondences and the functions can be viewed as Schur class operator functions on these balls.We will provide evidence to show that the elements in these (non commutative) Hardy algebras behave very much like bounded analytic functions and the study of these algebras should be viewed as noncommutative function theory.展开更多
文摘In this work,we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=∈_(ji)~kθ_k and a momentum noncommutativity matrix parameter β_(ij)=∈_(ij)~kβ_k,is shown to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS).Imposing some constraints on this particular transformation,we firstly find that the product of the two parameters θ and β possesses a lower bound in direct relation with Heisenberg incertitude relations,and secondly that the two parameters are equivalent but with opposite sign,up to a dimension factor depending on the physical system under study.This means that noncommutativity is represented by a unique parameter which may play the role of a fundamental constant characterizing the whole NCQPS.Within our framework,we treat some physical systems on NCQPS:free particle,harmonic oscillator,system of two-charged particles,Hydrogen atom.Among the obtained results, we discover a new phenomenon which consists of a free particle on NCQPS viewed as equivalent to a harmonic oscillator with Larmor frequency depending on β,representing the same particle in presence of a magnetic field=q~(-1).For the other examples,additional correction terms depending on β appear in the expression of the energy spectrum.Finally,in the two-particle system case,we emphasize the fact that for two opposite charges noncornmutativity is effectively feeled with opposite sign.
基金supported by National Natural Science Foundation of China(Grant Nos.11026046,11101179,10971071)Doctoral Fund of Ministry of Education of China(Grant No.20100061120096)the Fundamental Research Funds for the Central Universities(Grant No.200903294)
文摘In this paper,we give the notion of derivations of Lie 2-algebras using explicit formulas,and construct the associated derivation Lie 3-algebra.We prove that isomorphism classes of non-abelian extensions of Lie 2-algebras are classified by equivalence classes of morphisms from a Lie 2-algebra to a derivation Lie 3-algebra.
基金Project supported by the grant No. 1999-2-102-001-3 from the Interdisciplinary Research Program Year of the KOSEF
文摘Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aω of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr（Aω） = tr（Aρ）. It is proved that the set of all C*-algebras of sections of locally trivial C*-algebra bundles over S2 with fibres Aω. has a group structure, denoted by π1（Aut（Aω.））, which is isomorphic to Z if d > 1 and {0} if d > 1. Let Bcd be a cd-homogeneous C*-algebra over S2 x T2 of which no non-trivial matrix algebra can be factored out. The spherical noncommutative torns Sρcd is defined by twisting C*（T2 x Zm-2） in Bcd C* （Z（m-2）） by a totally skew multiplier ρ on T2 x Z（m-2）. It is shown that Sρcd Mp∞ is isomorphic to C（S2） C* （T2 x Zm-2, ρ） Mcd（C） Mp∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.
基金supported by National Natural Science Foundation of China (Grant Nos. 10725105, 10731080, 11021091)and Chinese Academy of Sciences
文摘This short paper is based on the talk on the conference Operator Algebras and Related Topics held on July 23-27, 2010, Beijing. The author surveys recent developments of the noncommutative gravity in joint works with Chaichian, Tureanu, Sun, Wang, Xie and Zhang.
基金Project supported by Grant No.1999-2-102-001-3 from the Interdisciplinary Research Program Year of the KOSEF.
文摘The generalized noncommutative torus Tkp of rank n was defined in [4] by the crossed product Am/k ×a3 Z ×a4 … ×an Z, where the actions ai of Z on the fibre Mk(C) of a rational rotation algebra Am/k are trivial, and C*(kZ × kZ) ×a3 Z ×a4 ... ×an Z is a completely irrational noncommutative torus Ap of rank n. It is shown in this paper that Tkp is strongly Morita equivalent to Ap, and that Tkp (?) Mp∞ is isomorphic to Ap (?) Mk(C) (?) Mp∞ if and only if the set of prime factors of k is a subset of the set of prime factors of p.
基金supported by a grant from the U.S.-Israel Binational Science Foundation (Grant No. 200641)supported by the Technion V.P.R. Fund
文摘In this expository paper,we describe the study of certain non-self-adjoint operator algebras,the Hardy algebras,and their representation theory.We view these algebras as algebras of (operator valued) functions on their spaces of representations.We will show that these spaces of representations can be parameterized as unit balls of certain W*-correspondences and the functions can be viewed as Schur class operator functions on these balls.We will provide evidence to show that the elements in these (non commutative) Hardy algebras behave very much like bounded analytic functions and the study of these algebras should be viewed as noncommutative function theory.
