Real spectra of quantum groups

Abstract

Abstract. The only noncommutative ring for which the real spectrum has been described so far is the quantum affine ring R q [x, y], see [21]. The aim of this paper is to describe the real spectra of quantum affine rings k q [x 1 , . . . , x n ] where k is a formally real affine R-algebra and q ∈ M n (R + ). As a by-product we describe the real spectra of quantized enveloping algebra U q (sl 2 (R)) and quantum special linear group O q (SL 2 (R)). 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.