On the Galois correspondence for Hopf Galois structures arising from finite radical algebras and Zappa-Szép products.

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Lindsay N. Childs
University at Albany (Albany. NY. Estats Units d'Àmerica). Department of Mathematics and Statistics

Let L/K be a G-Galois extension of fields with an H-Hopf Galois structure of type N. We study the Galois correspondence ratio GC(G, N), which is the proportion of intermediate fields E with K ⊆ E ⊆ L that are in the image of the Galois correspondence for the H-Hopf Galois structure on L/K. The Galois correspondence ratio for a Hopf Galois structure can be found by translating the problem into counting certain subgroups of the corresponding skew brace. We look at skew braces arising from finite radical algebras A and from Zappa–Sz´ep products of finite groups, and in particular when A3 = 0 or the Zappa–Sz´ep product is a semidirect product, in which cases the corresponding skew brace is a bi-skew brace, that is, a set G with two group operations ◦ and ? in such a way that G is a skew brace with either group structure acting as the additive group of the skew brace. We obtain the Galois correspondence ratio for several examples. In particular, if (G, ◦, ?) is a biskew brace of squarefree order 2m where (G, ◦) ∼= Z2m is cyclic and (G, ?) ∼= Dm is dihedral, then for large m, GC(Z2m, Dm) is close to 1/2 while GC(Dm, Z2m) is near 0.

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Galois correspondence, Hopf Galois extension, skew brace

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Childs, Lindsay N. “On the Galois correspondence for Hopf Galois structures arising from finite radical algebras and Zappa-Szép products”. Publicacions Matemàtiques, vol.VOL 65, no. 1, pp. 141–163, https://raco.cat/index.php/PublicacionsMatematiques/article/view/383689.
Referències
A. A. Alabdali and N. P. Byott, Counting Hopf–Galois structures on cyclic field extensions of squarefree degree, J. Algebra 493 (2018), 1–19. DOI: 10.1016/ j.jalgebra.2017.09.009.

N. P. Byott, Solubility criteria for Hopf–Galois structures, New York J. Math. 21 (2015), 883–903.

N. P. Byott and L. N. Childs, Fixed-point free pairs of homomorphisms and nonabelian Hopf–Galois structures, New York J. Math. 18 (2012), 707–731.

A. Caranti, F. Dalla Volta, and M. Sala, Abelian regular subgroups of the affine group and radical rings, Publ. Math. Debrecen 69(3) (2006), 297–308.

S. U. Chase and M. E. Sweedler, “Hopf Algebras and Galois Theory”, Lecture Notes in Mathematics 97, Springer-Verlag, Berlin-New York, 1969. DOI: 10. 1007/BFb0101433.

L. N. Childs, On the Galois correspondence for Hopf Galois structures, New York J. Math. 23 (2017), 1–10.

L. N. Childs, Skew braces and the Galois correspondence for Hopf Galois structures, J. Algebra 511 (2018), 270–291. DOI: 10.1016/j.jalgebra.2018.06.023.

L. N. Childs, Bi-skew braces and Hopf Galois structures, New York J. Math. 25 (2019), 574–588.

L. N. Childs and J. Corradino, Cayley’s Theorem and Hopf Galois structures for semidirect products of cyclic groups, J. Algebra 308(1) (2007), 236–251. DOI: 10.1016/j.jalgebra.2006.09.016.

L. N. Childs and C. Greither, Bounds on the number of ideals in finite commutative nilpotent Fp-algebras, Publ. Math. Debrecen 92(3–4) (2018), 495–516.

T. Crespo, A. Rio, and M. Vela, On the Galois correspondence theorem in separable Hopf Galois theory, Publ. Mat. 60(1) (2016), 221–234. DOI: 10.5565/PUBLMAT−60116−08.

K. De Commer, Actions of skew braces and set-theoretic solutions of the reflection equation, Proc. Edinb. Math. Soc. (2) 62(4) (2019), 1089–1113. DOI: 10.1017/s0013091519000129.

W. A. De Graaf, Classification of nilpotent associative algebras of small dimension, Internat. J. Algebra Comput. 28(1) (2018), 133–161. DOI: 10.1142/ S0218196718500078.

D. S. Dummit and R. M. Foote, “Abstract Algebra”, 2nd edition, John Wiley and Sons, New York, 1999.

S. C. Featherstonhaugh, A. Caranti, and L. N. Childs, Abelian Hopf Galois structures on prime-power Galois field extensions, Trans. Amer. Math. Soc. 364(7) (2012), 3675–3684. DOI: 10.1090/S0002-9947-2012-05503-6.

C. Greither and B. Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106(1) (1987), 239–258. DOI: 10.1016/0021-8693(87)90029-9.

L. Guarnieri and L. Vendramin, Skew braces and the Yang–Baxter equation, Math. Comp. 86(307) (2017), 2519–2534. DOI: 10.1090/mcom/3161.

I. N. Herstein, “Theory of Rings”, Mathematics Lecture Notes, University of Chicago, 1961.

A. Koch and P. J. Truman, Opposite skew left braces and applications, J. Algebra 546 (2020), 218–235. DOI: 10.1016/j.jalgebra.2019.10.033.

K. Nejabati Zenouz, Skew braces and Hopf–Galois structures of Heisenberg type, J. Algebra 524 (2019), 187–225. DOI: 10.1016/j.jalgebra.2019.01.012.

W. Rump, Braces, radical rings, and the quantum Yang–Baxter equation, J. Algebra 307(1) (2007), 153–170. DOI: 10.1016/j.jalgebra.2006.03.040.

A. Smoktunowicz and L. Vendramin, On skew braces (with an appendix by N. Byott and L. Vendramin), J. Comb. Algebra 2(1) (2018), 47–86. DOI: 10. 4171/JCA/2-1-3.