Real spectra of quantum groups

The only noncommutative ring for which the real spectrum has been described so far is the quantum affine ring [J. Algebra 212 (1999) 190]. The aim of this paper is to describe the real spectra of quantum affine rings kq[x1,…,xn] where k is a formally real affine -algebra and . As a by-product we describe the real spectra of quantized enveloping algebra and quantum special linear group . Formal reality and semireality is characterized for the following classes of quantum groups: quantum affine rings, quantized enveloping algebras, quantized function algebras, quantized Weyl algebras.