Real spectra of quantum groups (Q1879645)

Let \(R\) be a ring. The real spectrum of \(R\), introduced by \textit{K. H. Leung, M. Marshall} and \textit{Y. Zhang} in [J. Algebra 198, No. 2, 412--427 (1997; Zbl 0909.16027)] is by the definition the set of all orderings of \(R\). This paper describes the real spectra of quantum affine rings \(k_{\mathbf q}[x_1, \dots, x_n]\), where \(k\) is a formally real affine \(\mathbb R\)-algebra and \({\mathbf q} \in M_n(\mathbb R^+)\). The description of this real spectra is reduced to the description of the real spectra of polynomial rings. For \(k = \mathbb R\), these last description is given in turn in the paper cited above by Leung, Marshall and Zhang. The paper also gives a characterization of formal reality and semireality of quantum affine rings, quantized enveloping algebras, quantized function algebras and quantized Weyl algebras.