On the construction of unitary quantum group differential calculus∗∗This work has been funded by the Russian Academic Excellence Project ‘5-100’. The results of section (propositions 5.2, 5.3 and theorem 5.5) have been obtained under support of the RSF grant No.16-11-10160.

Abstract

We develop a construction of the unitary type anti-involution for the quantized differential calculus over in the case . To this end, we consider a joint associative algebra of quantized functions, differential forms and Lie derivatives over , which is bicovariant with respect to coactions. We define a specific non-central spectral extension of this algebra by the spectral variables of three matrices of the algebra generators. In the spectrally expended algebra, we construct a three-parametric family of its inner automorphisms. These automorphisms are used for the construction of the unitary anti-involution for the (spectrally extended) calculus over .