Title: On Hopf algebroid structure of kappa-deformed Heisenberg algebra Show Chinese title

Title: 关于κ-变形Heisenberg代数的Hopf代数结构

Abstract: 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 Poincar\'e 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. Show Chinese abstract

Abstract: $(4+4)$维$\kappa$变形量子相空间及其由洛伦兹 sector 所描述的$(10+10)$维协变扩展可以被描述为海森堡 doubles：$(10+10)$维量子相空间是$D=4$ $\kappa$ 变形Poincaré Hopf代数$\mathbb{H}$的 double，而标准的$(4+4)$维空间是由$\kappa$-Minkowski 坐标$\hat{x}_\mu$和相应的对易动量$\hat{p}_\mu$所生成的子代数。 每一个Heisenberg双积都表现为一个以基代数为底的Hopf代数组的总代数，在我们的例子中，这个基代数是坐标部分。 我们展示了这个结构的细节，即相应的右bialgebroid和反极性映射。 我们依赖于Majid-Ruegg双交叉积基中的代数计算方法。 目标映射来源于J-H. Lu的一个公式。 余乘法取值于基上的双模张量积，这被表达为余乘法规范自由度的存在。