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Working paper

On construction of unitary quantum group differential calculus

We develop a construction of the unitary type anti-involution for the quantized differential calculus over GL_q(n) in the case |q|=1. To this end, we consider a joint associative algebra of quantized functions, differential forms and Lie derivatives over GL_q(n)/SL_q(n), which is bicovariant with respect to GL_q(n)/SL_q(n) coactions. We define a specific non-central {\em 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 construction of the unitary anti-involution for the (spectrally extended) calculus over GL_q(n).