Set-theoretic solutions of the pentagon equation. (English) Zbl 1482.16059

Colazzo, Ilaria; Jespers, Eric; Kubat, Łukasz

Commun. Math. Phys. 380, No. 2, 1003-1024 (2020).

A set-theoretic solution of the Pentagon Equation on a non-empty set \(S\) is a pair \((S,s)\), where \(s\colon S\times S\to S\times S\) is a mapping of the form \(s(x,y)=(x\cdot y,\theta_x(y))=(x\cdot y,x\ast y)\), such that \((S,\cdot)\) is a semigroup and for \(x,y,z\in S\) the following additional conditions are satisfied: \[\theta_x(y)\cdot \theta_{x\cdot y}(z)=\theta_x(y\cdot z)\quad \text{and}\quad \theta_{\theta_x(y)}\theta_{x\cdot y}=\theta_y.\] A solution \((S,s)\) is called bijective if \(s\) is a bijection and is involutive if \(s^2=id_{S\times S}\).
F. Catino et al. [Commun. Algebra 48, No. 1, 83–92 (2020; Zbl 1447.16034)] described all solutions \((S,s)\) of the Pentagon Equation for which \((S,\cdot)\) and \((S,\ast)\) are groups. F. Catino et al. [Semigroup Forum 101, No. 2, 259–284 (2020; Zbl 1508.20081)] obtained solutions of the Pentagon Equation on the matched product of two semigroups.
In this paper, bijective solutions of the Pentagon Equation are investigated. The main result shows that each involutive solution \((S,s)\) of the Pentagon Equation may be constructed on a product \(X\times A\times G\) of three non-empty sets, where \((A,+)\) and \((G,\circ)\) are two elementary abelian \(2\)-groups, \(\sigma\colon A\to Sym(X)\) and for \(x,y\in X\), \(a,b\in A\), \(g,h\in G\): \[s((x,a,g),(y,b,h))=((x,a,g\circ h),(\sigma_{a+b}\sigma_b^{-1}(y),a+b,h)).\]
In a finite case with \(|S|=2^n(2m+1)\), there are, up to isomorphism, exactly \(\binom{n+2}{2}\) involutive solutions of the Pentagon Equation defined on a set \(S\).
Similarly as for the Yang-Baxter Equation the notion of an irretractable involutive solution of the Pentagon Equation is defined. The authors prove that for each such solution \((S,s)\) there exists an elementary abelian \(2\)-group \((S,+)\) such that for every \(x,y\in S\), \(s(x,y)=(x,x+y)\). They also show that two irretractable involutive solutions are isomorphic if and only if they are of the same cardinality.
To study involutive non-degenerate set theoretic solutions of the Yang-Baxter equation many different algebraic structures corresponding to such solutions have been introduced [see, e.g., P. Etingof et al., Duke Math. J. 100, 169–209 (1999; Zbl 0969.81030)]. In this paper, the structure algebra associated with an involutive finite solution of the Pentagon Equation is considered. In particular, the authors show that such algebra is a finite extension of a free abelian submonoid and is a Noetherian algebra satisfying a polynomial identity.

Reviewer: Agata Pilitowska (Warszawa)

MSC:

16T25	Yang-Baxter equations
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory
16T05	Hopf algebras and their applications
17B37	Quantum groups (quantized enveloping algebras) and related deformations
46L67	Quantum groups (operator algebraic aspects)

Keywords:

pentagon equation; elementary abelian 2-groups

Citations:

Zbl 1447.16034; Zbl 0969.81030; Zbl 1508.20081

References:

[1]	Ananin, A.Z.: An intriguing story about representable algebras. In: Ring Theory 1989 (Ramat Gan and Jerusalem, 1988/1989), Israel Mathemaics Conference Proceedings, vol. 1, pp. 31-38. Weizmann, Jerusalem (1989)
[2]	Baaj, S.; Skandalis, G., Unitaires multiplicatifs et dualité pour les produits croisés de \(C^*\)-algèbres, Ann. Sci. École Norm. Sup. (4), 26, 4, 425-488 (1993) · Zbl 0804.46078
[3]	Baaj, S.; Skandalis, G., Unitaires multiplicatifs commutatifs, C. R. Math. Acad. Sci. Paris, 336, 4, 299-304 (2003) · Zbl 1028.46083
[4]	Bachiller, D., Solutions of the Yang-Baxter equation associated to skew left braces, with applications to racks, J. Knot Theory Ramif., 27, 8, 1850055 (2018) · Zbl 1443.16040
[5]	Bachiller, D.; Cedó, F.; Jespers, E., Solutions of the Yang-Baxter equation associated with a left brace, J. Algebra, 463, 80-102 (2016) · Zbl 1348.16027
[6]	Baxter, RJ, Eight-vertex model in lattice statistics, Phys. Rev. Lett., 26, 832-833 (1971)
[7]	Catino, F.; Mazzotta, M.; Miccoli, MM, Set-theoretical solutions of the pentagon equation on groups, Commun. Algebra, 48, 1, 83-92 (2020) · Zbl 1447.16034
[8]	Catino, F.; Mazzotta, M.; Stefanelli, P., Set-theoretical solutions of the Yang-Baxter and pentagon equations on semigroups, Semigroup Forum (2020) · Zbl 1508.20081
[9]	Cedó, F.; Jespers, E.; Okniński, J., Retractability of set theoretic solutions of the Yang-Baxter equation, Adv. Math., 224, 6, 2472-2484 (2010) · Zbl 1192.81202
[10]	Cedó, F.; Jespers, E.; Okniński, J., Braces and the Yang-Baxter equation, Commun. Math. Phys., 327, 1, 101-116 (2014) · Zbl 1287.81062
[11]	Cedó, F.; Okniński, J., Gröbner bases for quadratic algebras of skew type, Proc. Edinb. Math. Soc. (2), 55, 2, 387-401 (2012) · Zbl 1348.20062
[12]	Clifford, AH; Preston, GB, The Algebraic Theory of semigroups (1961), Providence, RI: American Mathematical Society, Providence, RI
[13]	Dimakis, A.; Müller-Hoissen, F., Simplex and polygon equations, SIGMA Symmetry Integr. Geom. Methods Appl., 11, 49 (2015) · Zbl 1338.06001
[14]	Drinfeld, V.G.: On some unsolved problems in quantum group theory. In: Quantum Groups (Leningrad, 1990), Lecture Notes in Mathematics, vol. 1510, pp. 1-8. Springer, Berlin (1992)
[15]	Etingof, P.; Schedler, T.; Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100, 2, 169-209 (1999) · Zbl 0969.81030
[16]	Gateva-Ivanova, T., Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity, Adv. Math., 230, 4-6, 2152-2175 (2012) · Zbl 1267.81209
[17]	Gateva-Ivanova, T., Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups, Adv. Math., 338, 649-701 (2018) · Zbl 1437.16028
[18]	Gateva-Ivanova, T.; Van den Bergh, M., Semigroups of \(I\)-type, J. Algebra, 206, 1, 97-112 (1998) · Zbl 0944.20049
[19]	Guarnieri, L.; Vendramin, L., Skew braces and the Yang-Baxter equation, Math. Comput., 86, 307, 2519-2534 (2017) · Zbl 1371.16037
[20]	Jespers, E.; Kubat, Ł.; Van Antwerpen, A., The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang-Baxter equation, Trans. Am. Math. Soc., 372, 10, 7191-7223 (2019) · Zbl 1432.16032
[21]	Jespers, E.; Kubat, Ł.; Van Antwerpen, A., Corrigendum and addendum to “The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang-Baxter equation”, Trans. Am. Math. Soc., 373, 6, 4517-4521 (2020) · Zbl 1442.16036
[22]	Jespers, E.; Okniński, J., Noetherian Semigroup Algebras, Algebraic Applications (2007), Dordrecht: Springer, Dordrecht · Zbl 1178.16025
[23]	Jiang, L.; Liu, M., On set-theoretical solution of the pentagon equation, Adv. Math. (China), 34, 3, 331-337 (2005)
[24]	Kashaev, RM, The Heisenberg double and the pentagon relation, Algebra i Analiz, 8, 4, 63-74 (1996) · Zbl 0870.16023
[25]	Kashaev, R.M.: Fully noncommutative discrete Liouville equation. In: Infinite Analysis 2010. Developments in quantum integrable systems, RIMS Kôkyûroku Bessatsu, B28, pp. 89-98. Research Institute for Mathematical Sciences (RIMS), Kyoto (2011) · Zbl 1260.81075
[26]	Kashaev, R.M., Reshetikhin, N.Y.: Symmetrically factorizable groups and self-theoretical solutions of the pentagon equation. In: Quantum Groups, Contemporary Mathematics, vol. 433, pp. 267-279. American Mathematical Society, Providence, RI (2007) · Zbl 1177.17014
[27]	Kashaev, RM; Sergeev, SM, On pentagon, ten-term, and tetrahedron relations, Commun. Math. Phys., 195, 2, 309-319 (1998) · Zbl 0937.16045
[28]	Kassel, C., Quantum Groups (1995), New York: Springer, New York
[29]	Lu, J-H; Yan, M.; Zhu, Y-C, On the set-theoretical Yang-Baxter equation, Duke Math. J., 104, 1, 1-18 (2000) · Zbl 0960.16043
[30]	Maillet, JM, On pentagon and tetrahedron equations, Algebra i Analiz, 6, 2, 206-214 (1994) · Zbl 0815.58008
[31]	McConnell, J.C., Robson, J.C. (with the cooperation of Small, L.W.): Noncommutative Noetherian Rings, Graduate Studies in Mathematics, vol. 30, revised ed. American Mathematical Society, Providence, RI (2001) · Zbl 0980.16019
[32]	Militaru, G., The Hopf modules category and the Hopf equation, Commun. Algebra, 26, 10, 3071-3097 (1998) · Zbl 0907.16018
[33]	Militaru, G., Heisenberg double, pentagon equation, structure and classification of finite-dimensional Hopf algebras, J. Lond. Math. Soc. (2), 69, 1, 44-64 (2004) · Zbl 1063.16046
[34]	Okniński, J., Semigroup Algebras (1991), New York: Marcel Dekker Inc, New York
[35]	Polishchuk, A.; Positselski, L., Quadratic Algebras (2005), Providence, RI: American Mathematical Society, Providence, RI
[36]	Rump, W., A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math., 193, 1, 40-55 (2005) · Zbl 1074.81036
[37]	Street, R., Fusion operators and cocycloids in monoidal categories, Appl. Categ. Struct., 6, 2, 177-191 (1998) · Zbl 0912.16020
[38]	Yang, CN, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett., 19, 1312-1315 (1967) · Zbl 0152.46301
[39]	Zamolodchikov, AB, Tetrahedra equations and integrable systems in three-dimensional space, Sov. Phys. JETP, 52, 2, 325-336 (1980)
[40]	Zamolodchikov, AB, Tetrahedron equations and the relativistic \(S\)-matrix of straight-strings in \(2+1\)-dimensions, Commun. Math. Phys., 79, 4, 489-505 (1981)