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Register a new userTwo $theta_{mu u }$ -deformed covariant relativistic quantum phase spaces as Poincare-Hopf algebroids
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We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known $theta_{mu u }$-deformed $D=4$ Poincare-Hopf algebra $mathbb{H}$. The first algebroid describes $theta_{mu u }$-deformed relativistic phase space with canonical NC space-time (constant $theta_{mu u }$ parameters) and the second one incorporates dual to $mathbb{H}$ quantum $theta_{mu u }$-deformed Poincare-Hopf group algebra $mathbb{G}$, which contains noncommutative space-time translations given by $Lambda $-dependent $Theta_{mu u }$ parameters ($% Lambda $ $equiv Lambda_{mu u }$ parametrize classical Lorentz group). The canonical $theta_{mu u }$-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincare transformations described by $mathbb{G}$. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.
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We consider the general D=4 (10+10)-dimensional kappa-deformed quantum phase space as given by Heisenberg double mathcal{H} of D=4 kappa-deformed Poincare-Hopf algebra H. The standard (4+4) -dimensional kappa - deformed covariant quantum phase space spanned by kappa - deformed Minkowski coordinates and commuting momenta generators ({x}_{mu },{p}_{mu }) is obtained as the subalgebra of mathcal{H}. We study further the property that Heisenberg double defines particular quantum spaces with Hopf algebroid structure. We calculate by using purely algebraic methods the explicite Hopf algebroid structure of standard kappa - deformed quantum covariant phase space in Majid-Ruegg bicrossproduct basis. The coproducts for Hopf algebroids are not unique, determined modulo the coproduct gauge freedom. Finally we consider the interpretation of the algebraic description of quantum phase spaces as Hopf algebroids.
We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the considered generalized phase spaces into the ones describing the quantum-deformed Heisenberg algebras.
We consider the generalized (10+10)-dimensional D=4 quantum phase spaces containing translational and Lorentz spin sectors associated with the dual pair of twist-quantized Poincare Hopf algebra $mathbb{H}$ and quantum Poincare Hopf group $widehat{mathbb{G}}$. Two Hopf algebroid structures of generalized phase spaces with spin sector will be investigated: first one $% mathcal{H}^{(10,10)}$ describing dynamics on quantum group algebra $% widehat{mathbb{G}}$ provided by the Heisenberg double algebra $mathcal{HD=% }mathbb{H}rtimes widehat{mathbb{G}}$, and second, denoted by $mathcal{% tilde{H}}^{(10,10)}$, describing twisted Hopf algebroid with base space containing twisted noncommutative Minkowski space $hat{x}_{mu }$. We obtain the first explicit example of Hopf algebroid structure of relativistic quantum phase space which contains quantum-deformed Lorentz spin sector.
The Minkowski spacetime quantum Clifford algebra structure associated with the conformal group and the Clifford-Hopf alternative k-deformed quantum Poincare algebra is investigated in the Atiyah-Bott-Shapiro mod 8 theorem context. The resulting algebra is equivalent to the deformed anti-de Sitter algebra U_q(so(3,2)), when the associated Clifford-Hopf algebra is taken into account, together with the associated quantum Clifford algebra and a (not braided) deformation of the periodicity Atiyah-Bott-Shapiro theorem.
The classical $r$-matrix for $N=1$ superPoincar{e} algebra, given by Lukierski, Nowicki and Sobczyk is used to describe the graded Poisson structure on the $N=1$ Poincar{e} supergroup. The standard correspondence principle between the even (odd) Poisson brackets and (anti)commutators leads to the consistent quantum deformation of the superPoincar{e} group with the deformation parameter $q$ described by fundamental mass parameter $kappa quad (kappa^{-1}=ln{q})$. The $kappa$-deformation of $N=1$ superspace as dual to the $kappa$-deformed supersymmetry algebra is discussed.
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