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An involution and dynamics for the $q$-deformed quantum top

A. Yu. Alekseev·L. D. Faddeev
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Abstract

It is known that the involution corresponding to the compact form is incompatible with comultiplication for quantum groups at $|q|=1$. In this paper we consider the quantum algebra of functions on the deformed space $T^{*}G_{q}$ which includes both the quantum group and the quantum universal enveloping algebra as subalgebras. For this extended object we construct an anti-involution which reduces to the compact form $*$-operation in the limit $q\rightarrow 1$. The algebra of functions on $T^{*}G_{q}$ endowed with the $*$-operation may be viewed as an algebra of observables of a quantum mechanical system. The most natural interpretation for such a system is a deformation of the quantum symmetric top. We suggest a discrete dynamics for this system which imitates the free motion of the top.

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A. Yu. Alekseev, L. D. Faddeev. 1994-06-29. An involution and dynamics for the $q$-deformed quantum top. https://arxiv.org/abs/hep-th/9406196

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