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The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang-Baxter equation. (English) Zbl 1432.16032

Trans. Am. Math. Soc. 372, No. 10, 7191-7223 (2019); corrigendum ibid. 373, No. 6, 4517-4521 (2020).
Let \((X,r)\) be a bijective left non-degenerate set-theoretic solution of the Yang-Baxter equation, and let \(M(X,r)\) be the associated structure monoid, defined in [T. Gateva-Ivanova and M. Van den Bergh, J. Algebra 206, No. 1, 97–112 (1998; Zbl 0944.20049)] and also in [P. Etingof et al., Duke Math. J. 100, No. 2, 169–209 (1999; Zbl 0969.81030)]. The aim of the paper under review is to study \(M(X,r)\) and its monoid algebra \(K[M(X,r)]\), called the structure algebra of the solution. The authors define another monoid \(A(X,r)\), called the derived structure monoid, which turns out to be the structure monoid of a rack, and they show that \(M(X,r)\) is a regular submonoid of a semidirect product \(A(X,r)\rtimes \mathrm{Sym}(X)\). If the solution \((X,r)\) is finite, then both \(A(X,r)\) and \(M(X,r)\) are central-by-finite monoids, and their monoid algebras \(K[A(X,r)]\) and \(K[M(X,r)]\) over a field \(K\) are Noetherian and PI, and moreover these algebras have the same classical Krull dimension, and the same Gelfand-Kirillov dimension, all these dimensions being equal to the rank of \(M(X,r)\) (and also to the rank of \(A(X,r)\)). A conjecture of Gateva-Ivanova is solved by showing that \(M(X,r)\) is cancellative if and only if \((X,r)\) is involutive. The prime ideals of the monoids \(M(X,r)\) and \(A(X,r)\), and of the algebras \(K[A(X,r)]\) and \(K[M(X,r)]\) are investigated. If \(P\) is a prime ideal of \(K[M(X,r)]\), a matrix representation of the algebra \(K[M(X,r)]/P\) is given. As a consequence, it is proved that if \(K[M(X,r)]\) is semiprime, then it embeds in a finite product of full matrix algebras over the group algebras of certain finitely generated abelian-by-finite groups, each of these groups being the group of quotients of a cancellative subsemigroup of \(M(X,r)\).

MSC:

16T25 Yang-Baxter equations
16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings
16N60 Prime and semiprime associative rings
16S36 Ordinary and skew polynomial rings and semigroup rings
16S37 Quadratic and Koszul algebras
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References:

[1] Amberg, B.; Dickenschied, O.; Sysak, Ya. P., Subgroups of the adjoint group of a radical ring, Canad. J. Math., 50, 1, 3-15 (1998) · Zbl 0908.20031 · doi:10.4153/CJM-1998-001-9
[2] Anan\cprime in, A. Z., An intriguing story about representable algebras. Ring theory 1989, Ramat Gan and Jerusalem, 1988/1989, Israel Math. Conf. Proc. 1, 31-38 (1989), Weizmann, Jerusalem · Zbl 0682.16013
[3] Bachiller, David, Solutions of the Yang-Baxter equation associated to skew left braces, with applications to racks, J. Knot Theory Ramifications, 27, 8, 1850055, 36 pp. (2018) · Zbl 1443.16040 · doi:10.1142/S0218216518500554
[4] Bachiller, David, Extensions, matched products, and simple braces, J. Pure Appl. Algebra, 222, 7, 1670-1691 (2018) · Zbl 1437.20031 · doi:10.1016/j.jpaa.2017.07.017
[5] Bachiller, David, Classification of braces of order \(p^3\), J. Pure Appl. Algebra, 219, 8, 3568-3603 (2015) · Zbl 1312.81099 · doi:10.1016/j.jpaa.2014.12.013
[6] Bachiller, David; Ced\'{o}, Ferran; Jespers, Eric, Solutions of the Yang-Baxter equation associated with a left brace, J. Algebra, 463, 80-102 (2016) · Zbl 1348.16027 · doi:10.1016/j.jalgebra.2016.05.024
[7] BCJO D. Bachiller, F. Ced\'o, E. Jespers, and J. Okni\'nski, Asymmetric product of left braces and simplicity; new solutions of the Yang-Baxter equation, Commun. Contemp. Math. (2018), 1850042, https://doi.org/10.1142/S0219199718500426doi.org/10.1142/S0219199718500426. · Zbl 1451.16029
[8] Baxter, Rodney J., Partition function of the eight-vertex lattice model, Ann. Physics, 70, 193-228 (1972) · Zbl 0236.60070 · doi:10.1016/0003-4916(72)90335-1
[9] Braun, Amiram, The nilpotency of the radical in a finitely generated PI ring, J. Algebra, 89, 2, 375-396 (1984) · Zbl 0538.16013 · doi:10.1016/0021-8693(84)90224-2
[10] Catino, Francesco; Colazzo, Ilaria; Stefanelli, Paola, Semi-braces and the Yang-Baxter equation, J. Algebra, 483, 163-187 (2017) · Zbl 1385.16035 · doi:10.1016/j.jalgebra.2017.03.035
[11] Ced\'{o}, Ferran, Left braces: solutions of the Yang-Baxter equation, Adv. Group Theory Appl., 5, 33-90 (2018) · Zbl 1403.16033
[12] Ced\'{o}, Ferran; Gateva-Ivanova, Tatiana; Smoktunowicz, Agata, On the Yang-Baxter equation and left nilpotent left braces, J. Pure Appl. Algebra, 221, 4, 751-756 (2017) · Zbl 1397.16033 · doi:10.1016/j.jpaa.2016.07.014
[13] Ced\'{o}, Ferran; Jespers, Eric; Okni\'{n}ski, Jan, Braces and the Yang-Baxter equation, Comm. Math. Phys., 327, 1, 101-116 (2014) · Zbl 1287.81062 · doi:10.1007/s00220-014-1935-y
[14] Clifford, A. H.; Preston, G. B., The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, xv+224 pp. (1961), American Mathematical Society, Providence, R.I. · Zbl 0111.03403
[15] Drinfel\cprime d, V. G., On some unsolved problems in quantum group theory. Quantum groups, Leningrad, 1990, Lecture Notes in Math. 1510, 1-8 (1992), Springer, Berlin · Zbl 0765.17014 · doi:10.1007/BFb0101175
[16] Etingof, Pavel; Schedler, Travis; Soloviev, Alexandre, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100, 2, 169-209 (1999) · Zbl 0969.81030 · doi:10.1215/S0012-7094-99-10007-X
[17] Faith, Carl, Algebra. II, xviii+302 pp. (1976), Springer-Verlag, Berlin-New York · Zbl 0335.16002
[18] Gateva-Ivanova, Tatiana, Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups, Adv. Math., 338, 649-701 (2018) · Zbl 1437.16028 · doi:10.1016/j.aim.2018.09.005
[19] Ga2 T. Gateva-Ivanova, A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equation, preprint, https://arxiv.org/abs/1808.03938arXiv:1808.03938, 2018.
[20] Gateva-Ivanova, T.; Jespers, Eric; Okni\'{n}ski, Jan, Quadratic algebras of skew type and the underlying monoids, J. Algebra, 270, 2, 635-659 (2003) · Zbl 1054.16024 · doi:10.1016/j.jalgebra.2003.06.005
[21] Gateva-Ivanova, Tatiana; Van den Bergh, Michel, Semigroups of \(I\)-type, J. Algebra, 206, 1, 97-112 (1998) · Zbl 0944.20049 · doi:10.1006/jabr.1997.7399
[22] Goffa, Isabel; Jespers, Eric, Monoids of IG-type and maximal orders, J. Algebra, 308, 1, 44-62 (2007) · Zbl 1121.16025 · doi:10.1016/j.jalgebra.2006.07.029
[23] Guarnieri, L.; Vendramin, L., Skew braces and the Yang-Baxter equation, Math. Comp., 86, 307, 2519-2534 (2017) · Zbl 1371.16037 · doi:10.1090/mcom/3161
[24] Jespers, Eric; Okni\'{n}ski, Jan, Noetherian semigroup algebras, Algebra and Applications 7, x+361 pp. (2007), Springer, Dordrecht · Zbl 1135.16001
[25] Jespers, Eric; Okni\'{n}ski, Jan, Monoids and groups of \(I\)-type, Algebr. Represent. Theory, 8, 5, 709-729 (2005) · Zbl 1091.20024 · doi:10.1007/s10468-005-0342-7
[26] Jespers, Eric; Okni\'{n}ski, Jan; Van Campenhout, Maya, Finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids, J. Algebra, 440, 72-99 (2015) · Zbl 1346.16024 · doi:10.1016/j.jalgebra.2015.05.017
[27] Jespers, Eric; Van Antwerpen, Arne, Left semi-braces and solutions of the Yang-Baxter equation, Forum Math., 31, 1, 241-263 (2019) · Zbl 1456.16035 · doi:10.1515/forum-2018-0059
[28] Jespers, Eric; Van Campenhout, Maya, Finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids II, J. Algebra, 492, 524-546 (2017) · Zbl 1379.16017 · doi:10.1016/j.jalgebra.2017.09.011
[29] Kamada, Seiichi, Knot invariants derived from quandles and racks. Invariants of knots and 3-manifolds, Kyoto, 2001, Geom. Topol. Monogr. 4, 103-117 (2002), Geom. Topol. Publ., Coventry · Zbl 1037.57005 · doi:10.2140/gtm.2002.4.103
[30] KSV A. Konovalov, A. Smoktunowicz, and L. Vendramin, On skew braces and their ideals, Exp. Math., online (2018), https://doi.org/10.1080/10586458.2018.1492476doi.org/10.1080/10586458.2018.1492476. · Zbl 1476.16036
[31] Lebed, Victoria, Cohomology of idempotent braidings with applications to factorizable monoids, Internat. J. Algebra Comput., 27, 4, 421-454 (2017) · Zbl 1380.16035 · doi:10.1142/S0218196717500229
[32] LV V. Lebed and L. Vendramin, On structure groups of set-theoretic solutions to the Yang-Baxter equation, Proc. Edinb. Math. Soc., online (2019), https:/doi.org/10.1017/S0013091518000548doi.org/10.1017/S0013091518000548. · Zbl 1423.16034
[33] Lebed, Victoria; Vendramin, Leandro, Cohomology and extensions of braces, Pacific J. Math., 284, 1, 191-212 (2016) · Zbl 1357.20009 · doi:10.2140/pjm.2016.284.191
[34] Lu, Jiang-Hua; Yan, Min; Zhu, Yong-Chang, On the set-theoretical Yang-Baxter equation, Duke Math. J., 104, 1, 1-18 (2000) · Zbl 0960.16043 · doi:10.1215/S0012-7094-00-10411-5
[35] McConnell, J. C.; Robson, J. C., Noncommutative Noetherian rings, Graduate Studies in Mathematics 30, xx+636 pp. (2001), American Mathematical Society, Providence, RI · Zbl 0980.16019 · doi:10.1090/gsm/030
[36] Okni\'{n}ski, Jan, Semigroups of matrices, Series in Algebra 6, xiv+311 pp. (1998), World Scientific Publishing Co., Inc., River Edge, NJ · Zbl 0911.20042 · doi:10.1142/9789812816290
[37] Okni\'{n}ski, Jan, Semigroup algebras, Monographs and Textbooks in Pure and Applied Mathematics 138, x+357 pp. (1991), Marcel Dekker, Inc., New York · Zbl 0725.16001
[38] Passman, Donald S., Infinite crossed products, Pure and Applied Mathematics 135, xii+468 pp. (1989), Academic Press, Inc., Boston, MA · Zbl 0662.16001
[39] Passman, Donald S., The algebraic structure of group rings, Pure and Applied Mathematics, xiv+720 pp. (1977), Wiley-Interscience [John Wiley & Sons], New York-London-Sydney · Zbl 0368.16003
[40] Roseblade, J. E., Prime ideals in group rings of polycyclic groups, Proc. London Math. Soc. (3), 36, 3, 385-447 (1978) · Zbl 0391.16008 · doi:10.1112/plms/s3-36.3.385
[41] Rump, Wolfgang, Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra, 307, 1, 153-170 (2007) · Zbl 1115.16022 · doi:10.1016/j.jalgebra.2006.03.040
[42] Rump, Wolfgang, Modules over braces, Algebra Discrete Math., 2, 127-137 (2006) · Zbl 1164.81328
[43] Rump, Wolfgang, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math., 193, 1, 40-55 (2005) · Zbl 1074.81036 · doi:10.1016/j.aim.2004.03.019
[44] Smoktunowicz, Agata, On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation, Trans. Amer. Math. Soc., 370, 9, 6535-6564 (2018) · Zbl 1440.16040 · doi:10.1090/tran/7179
[45] Smoktunowicz, Agata; Vendramin, Leandro, On skew braces (with an appendix by N. Byott and L. Vendramin), J. Comb. Algebra, 2, 1, 47-86 (2018) · Zbl 1416.16037 · doi:10.4171/JCA/2-1-3
[46] Soloviev, Alexander, Non-unitary set-theoretical solutions to the quantum Yang-Baxter equation, Math. Res. Lett., 7, 5-6, 577-596 (2000) · Zbl 1046.81054 · doi:10.4310/MRL.2000.v7.n5.a4
[47] Yang, C. N., 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 · doi:10.1103/PhysRevLett.19.1312
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