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Register a new userLie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach
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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.
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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.
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.
The $(4+4)$-dimensional $kappa$-deformed quantum phase space as well as its $(10+10)$-dimensional covariant extension by the Lorentz sector can be described as Heisenberg doubles: the $(10+10)$-dimensional quantum phase space is the double of $D=4$ $kappa$-deformed Poincare Hopf algebra $mathbb{H}$ and the standard $(4+4)$-dimensional space is its subalgebra generated by $kappa$-Minkowski coordinates $hat{x}_mu$ and corresponding commuting momenta $hat{p}_mu$. Every Heisenberg double appears as the total algebra of a Hopf algebroid over a base algebra which is in our case the coordinate sector. We exhibit the details of this structure, namely the corresponding right bialgebroid and the antipode map. We rely on algebraic methods of calculation in Majid-Ruegg bicrossproduct basis. The target map is derived from a formula by J-H. Lu. The coproduct takes values in the bimodule tensor product over a base, what is expressed as the presence of coproduct gauge freedom.
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.
We proceed to study Yang-Baxter deformations of 4D Minkowski spacetime based on a conformal embedding. We first revisit a Melvin background and argue a Lax pair by adopting a simple replacement law invented in 1509.00173. This argument enables us to deduce a general expression of Lax pair. Then the anticipated Lax pair is shown to work for arbitrary classical $r$-matrices with Poincae generators. As other examples, we present Lax pairs for pp-wave backgrounds, the Hashimoto-Sethi background, the Spradlin-Takayanagi-Volovich background.
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