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Cocommutative elements form a maximal commutative subalgebra in quantum matrices. (English) Zbl 1418.16020

Summary: In this paper, we prove that the subalgebras of cocommutative elements in the quantized coordinate rings of \(M_n\), \(\mathrm{GL}_n\) and \(\mathrm{SL}_n\) are the centralizers of the trace \(x_{1, 1} + \cdots + x_{n, n}\) in each algebra, for \(q \in \mathbb{C}^\times\) being not a root of unity. In particular, it is not only a commutative subalgebra as it was known before, but it is a maximal one.

MSC:

16T20 Ring-theoretic aspects of quantum groups
20G42 Quantum groups (quantized function algebras) and their representations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
13A50 Actions of groups on commutative rings; invariant theory
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