Einstein–Rosen waves with two polarizations are cylindrically symmetric solutions to vacuum Einstein equations. Einstein equations in this case reduce to an integrable system. In 1971, Geroch has shown that this system admits an infinite-dimensional group of symmetry transformations known as the Geroch group. The phase space of this system can be parametrized by a matrix-valued function of spectral parameter, called monodromy matrix. The latter admits the Riemann–Hilbert factorization into a pair of transition matrices, i.e. matrix-valued functions of spectral parameter such that one of them is holomorphic in the upper half-plane, and the other is holomorphic in the lower half-plane. The classical Geroch group preserves the determinants of transition and monodromy matrices by construction. The algebraic quantization of the quadratic Poisson algebra generated by transition matrices of Einstein–Rosen system was proposed by Korotkin and Samtleben in 1997. Paternain, Peraza and Reisenberger have recently suggested a quantization of the Geroch group, which can be considered as a symmetry of the quantum algebra of observables. They have shown that commutation relations involving quantum monodromy matrices are preserved by the action of the quantum Geroch group. In present paper we introduce the notion of the determinant of the quantum monodromy matrix. We derive a factorization formula expressing the quantum determinant of the monodromy matrix as a product of the quantum determinants of the transition matrices. The action of the quantum Geroch group is extended from the subalgebra generated by the monodromy matrixonto the full algebra of observables. This extension is used to prove that the quantum determinant of the quantum monodromy matrix is invariant under the action of quantum Geroch group.

具有两个极化的爱因斯坦-罗森波是真空爱因斯坦方程的圆柱对称解。在这种情况下，爱因斯坦方程简化为可积系统。1971年，格罗赫（Geroch）表明，该系统允许一个无限维的对称变换组，称为格罗赫（Geroch）组。该系统的相空间可以通过称为单峰矩阵的光谱参数矩阵值函数进行参数化。后者允许将Riemann-Hilbert分解分解为一对过渡矩阵，即光谱参数的矩阵值函数，以便其中一个在上半平面为全纯，而另一个在下半平面为全纯。经典的Geroch小组通过构造保留了过渡和单峰矩阵的决定因素。由Einstein-Rosen系统的转换矩阵生成的二次Poisson代数的代数量化由Korotkin和Samtleben于1997年提出。Paternain，Peraza和Reisenberger最近提出了对Geroch群的量化，这可以看作是对等对称性。可观测量的量子代数。他们表明，涉及量子单峰矩阵的换向关系通过量子Geroch基团的作用得以保留。在本文中，我们介绍了量子单峰矩阵行列式的概念。我们导出一个分解式，将单峰矩阵的量子行列式表示为过渡矩阵的量子行列式的乘积。量子Geroch基团的作用从单峰矩阵产生的子代数扩展到了可观测物的全部代数。该扩展用于证明量子单峰矩阵的量子行列式在量子Geroch基团的作用下是不变的。