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Hidden classical symmetry in quantum spaces at roots of unity : From qsphere to fuzzy sphere,” hepth/0008186
"... We study relations between different kinds of noncommutative spheres which have appeared in the context of ADS/CFT correspondences recently, emphasizing the connections between spaces that have manifest quantum group symmetry and spaces that have manifest classical symmetry. In particular we consid ..."
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We study relations between different kinds of noncommutative spheres which have appeared in the context of ADS/CFT correspondences recently, emphasizing the connections between spaces that have manifest quantum group symmetry and spaces that have manifest classical symmetry. In particular we consider the quotient SUq(2)/U(1) at roots of unity, and find its relations with the fuzzy sphere with manifest classical SU(2) symmetry. Deformation maps between classical and quantum symmetry, the Uq(SU(2)) module structure of quantum spheres and the structure of indecomposable representations of Uq(SU(2)) at roots of unity conspire in an interesting way to allow the relation between manifestly Uq(SU(2) symmetric spheres and manifestly U(SU(2)) symmetric spheres. The relation suggests that a subset of field theory actions on the qsphere are equivalent to actions on the fuzzy sphere. The results here are compatible with the proposal that quantum spheres at roots of unity appear as effective geometries which account for finite N effects in the ADS/CFT correspondence. July
Quantum Fields on the GroenewoldMoyal Plane
, 2008
"... We give an introductory review of quantum physics on the noncommutative spacetime called the GroenewoldMoyal plane. Basic ideas like star products, twisted statistics, second quantized fields and discrete symmetries are discussed. We also outline some of the recent developments in these fields and ..."
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We give an introductory review of quantum physics on the noncommutative spacetime called the GroenewoldMoyal plane. Basic ideas like star products, twisted statistics, second quantized fields and discrete symmetries are discussed. We also outline some of the recent developments in these fields and mention where one can search for experimental signals.
The Niels Bohr Institute,
, 805
"... We investigate the Hopf algebra structure in string worldsheet theory and give a unified formulation of the quantization of string and the spacetime symmetry. We reformulate the pathintegral quantization of string as a Drinfeld twist on the worldsheet level. The coboundary relation shows that it i ..."
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We investigate the Hopf algebra structure in string worldsheet theory and give a unified formulation of the quantization of string and the spacetime symmetry. We reformulate the pathintegral quantization of string as a Drinfeld twist on the worldsheet level. The coboundary relation shows that it is equivalent to operators with normal ordering. By the twist, spacetime diffeomorphism is deformed into a twisted Hopf algebra, while the Poincaré symmetry is unchanged. This suggest a characterization of the symmetry: unbroken symmetries are twist invariant Hopf subalgebras, while broken symmetries are realized as twisted ones. We give arguments to relate this twisted Hopf algebra with symmetries in pathintegral quantization. 1
Twist Quantization of String and B Field Background
, 811
"... Abstract: In a previous paper, we investigated the Hopf algebra structure in string theory and gave a unified formulation of the quantization of the string and the spacetime symmetry. In this paper, this formulation is applied to the case with a nonzero Bfield background, and the twist of the Poinc ..."
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Abstract: In a previous paper, we investigated the Hopf algebra structure in string theory and gave a unified formulation of the quantization of the string and the spacetime symmetry. In this paper, this formulation is applied to the case with a nonzero Bfield background, and the twist of the Poincaré symmetry is studied. The Drinfeld twist accompanied by the Bfield background gives an alternative quantization scheme, which requires a new normal ordering. In order to obtain a physical interpretation of this twisted Hopf algebra structure, we propose a method to decompose the twist into two successive twists and we give two different possibilities of decomposition. The first is a natural decomposition from the viewpoint of the twist quantization, leading to a new type of twisted Poincaré symmetry. The second decomposition reveals the relation of our formulation to the twisted Poincaré symmetry on the Moyal type noncommutative space. Contents
at roots of unity: From qsphere to fuzzy sphere.
, 2000
"... Hidden classical symmetry in quantum spaces ..."
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Twist Quantization of String and Hopf Algebraic Symmetry
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2010
"... We describe the twist quantization of string worldsheet theory, which unifies the description of quantization and the target space symmetry, based on the twisting of Hopf and module algebras. We formulate a method of decomposing a twist into successive twists to analyze the twisted Hopf and module ..."
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We describe the twist quantization of string worldsheet theory, which unifies the description of quantization and the target space symmetry, based on the twisting of Hopf and module algebras. We formulate a method of decomposing a twist into successive twists to analyze the twisted Hopf and module algebra structure, and apply it to several examples, including finite twisted diffeomorphism and extra treatment for zero modes.