The Coxeter relations and KP map for non-commuting symbols
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The Coxeter relations and KP map for non-commuting symbols Adam Doliwa1
· Masatoshi Noumi2
Received: 1 November 2019 / Revised: 1 November 2019 / Accepted: 16 July 2020 © The Author(s) 2020
Abstract We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Mal’cev–Newmann division ring. The action is constructed from the non-Abelian Hirota–Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case, and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages. Keywords Discrete integrable systems · Non-commutative Hirota–Miwa equation · Yang–Baxter maps · Symmetric group · Mal’cev–Neumann series · Context-free languages Mathematics Subject Classification 37K10 · 37K60 · 16T25 · 39A14 · 14E07 · 12E15
1 Introduction Non-commutative extensions of integrable systems are of growing interest in mathematical physics [6,13–15,18,24,27,37,41,50,52]. They may be considered as a useful platform to more thorough understanding of integrable quantum or statistical mechanics lattice systems, where the quantum Yang–Baxter equation [3,38] plays a role. Recently, integrable maps with variables satisfying various commutation relations
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Adam Doliwa [email protected] http://wmii.uwm.edu.pl/~doliwa/ Masatoshi Noumi [email protected]
1
Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Słoneczna 54, 10-710 Olsztyn, Poland
2
Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
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A. Doliwa, M. Noumi
(including the canonical Weyl relations and those coming from the quantum group theory) have been studied in [4,21,28,42,57,60]. Such a direction from commutative to non-commutative objects in the theory of integrable systems is not an isolated issue. One can say it is in mainstream of current developments in contemporary mathematics. The origins of this trend are rooted back in quantum physics, where the idea of replacing functions by not necessarily commuting operators appeared. There is a topology of non-commutative spaces, a non-commutative integration theory, a non-commutative differential geometry [11], non-commutative probability theory [63], etc. The present research aims to make a step in direction to create a kind of free integrability. The advantage of working with totally non-commutative variables has been advocated also in [26]. As the geometric research [14] suggests, the algebraic environment of our work will be the theory of Mal’cev–Neumann division rings constructed from free groups. In this paper, we study maps obeying the Coxeter relations, which are obtained from the non-commutative discrete KP system of equations (called originally in [52] the non-Abelian Hirota–Miwa system). In the commutative case, such maps were studied in [23,34,59], see also [17] for a version with certain commutativity restr
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