A basis for the right quantum algebra and the ``\(1= q\)'' principle
The authors develop an algebraic framework which clarifies the underlying setting of the quantum MacMahon Master identity from [\textit{S. Garoufalidis, T. T. Q. Lê}, and \textit{D. Zeilberger}, Proc. Natl. Acad. Sci. USA 103, No.~38, 13928--13931 (2006; Zbl 1170.05012)]. In particular, using Bergman's Diamond Lemma [\textit{G. M. Bergman}, Adv. Math. 29, 178--218 (1978; Zbl 0377.16013)], they show that the \(q\)-right quantum algebra \(R_q\) (depending on a parameter \(q\)) and the corresponding \(1\)-right quantum algebra \(R\) for \(q=1\) have as basis the same set of irreducible biwords (Theorems 1 and 2). As a consequence, they arrive at the ''\(1=q\)'' principle which states that certain type of identities modulo the defining relations of \(R_q\) have corresponding counterpart identities which hold modulo the same relations where \(q=1\) (Theorem 3). Using this machinery, they discover the so-called strong quantum MacMahon Master Theorem (Corollary 5).
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