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Skew left brace
In mathematics, a skew left brace $$B$$is a set with two operations, $$\cdot$$ and $$\circ$$  so that $$B$$with the operation $$\cdot$$ is a group, often called the additive group, $$B$$with the operation  $$\circ$$  is a group, often called the circle group, and a certain compatibility condition, analogous to distributivity, connects the two group operations. First defined in 2017, skew left braces yield set-theoretic solutions of the Yang-Baxter equation (See Yang-Baxter equation) and also have a close connection with Hopf-Galois structures on classical Galois extensions of fields.

Definitions
A skew left brace is a set $$'B$$ with two operations, $$\cdot$$ and $$\circ$$, somewhat analogous to a ring (see Ring (mathematics)). The set $$B$$ with the operation $$\cdot$$, denoted    $$(B, \cdot)$$,    is a group with identity element 1, often called the \emph{additive group}, $$B$$ with the operation $$\circ$$, denoted $$(B, \circ)$$,    is a group, often called the \emph{circle group}, and the compatibility condition

holds for all $$a$$, $$b$$, $$c$$' in $$B$$. Here $$a^{-1}$$ denotes the inverse of $$a$$ in the group $$(B, \cdot)$$, and $$ \overline{a}$$ will denote the inverse of $$a$$ in the group $$(B, \circ)$$.

The identity element $$I$$ of the group $$(B, \circ) $$ is equal to the identity element 1 of  $$(B, \cdot)$$:    for setting $$a = I, c= 1$$ in the compatibility condition shows that  $$b = b \cdot I^{-1} $$ for all $$b$$ in $$B$$, hence by uniqueness of identity in $$(B, \cdot)$$ ,     $$I^{-1} = 1$$, hence  $$I = 1^{-1} = 1$$. A skew left brace $$B$$ whose additive group $$(B, \cdot)$$ is abelian: $$ a \cdot b = b \cdot a$$ for all $$a, b$$ in $$B$$, is called a left brace. Examples. The most trivial example of a skew brace is a group $$G = (G, \cdot)$$  made into a skew brace $$(G, \cdot, \circ) $$ where  $$\cdot = \cdot = \circ$$  (see Group (mathematics)). Less trivial: if $$N$$ is a non-abelian group, then defining $$\circ$$  on $$N$$ by  $$a \circ b = b \cdot a$$  makes $$N$$ into a skew brace.

Radical ring.    A ring $$A$$ without unity is a set with two operations, $$+$$ (addition) and  $$\cdot $$ (multiplication), where $$a \cdot b$$ is typically written $$ab$$, and $$a \cdot a \cdot \ldots \cdot a$$ ($$n$$ factors) is denoted  $$a^n$$. With those operations $$A$$ satisfies all of the properties of a ring (associativity of multiplication, left and right distributivity of multiplication over addition, etc.) except that $$A$$ has no multiplicative identity element.
 * Given any ring $$A$$, the circle operation $$\circ $$ on $$A$$ is defined by $$a \circ b = a + b + a\cdot b$$.    It is easy to check that the operation $$\circ$$ is associative, and $$a \circ 0 = 0 \circ a = a$$, so $$(A, \circ)$$, the set $$A$$ with the circle operation $$\circ$$, is a monoid with identity element equal to the additive identity element 0 of the ring $$A$$:  (see monoid).

The ring $$A$$ is a radical ring if and only if $$(A, \circ)$$ is a group:    that is, given any $$a$$ in  $$A$$, there is some $$b in A so that : see Jacobson radical, where such an element is called right quasi-regular and left quasi-regular. Thus A is a radical ring if and only if A is equal to its Jacobson radical J(A). The group (A, \circ) is sometimes called the \emph{adjoint group} of A.  See Radical ring.

A radical ring is a skew left brace. If A is a radical ring, then with the operations + and \circ, A is a left brace. The defining property:    for all a, b, c in A, = becomes, after replacing x \circ y by x + y + xy for all x, y in A, and the two sides are equal if and only if the left distributive law  $$a(b+c) = ab + ac$$ holds.

$$A$$ is also a right brace, because the corresponding defining property for a right brace is equivalent to right distributivity. Conversely, suppose $$B$$ is a set with two operations $$ + $$ and $$\circ$$     which make $$B$$ into both a left brace and a right brace. Define $$\cdot$$  on $$B$$ by     $$x \cdot y = x \circ y - x - y$$. Then the set $$B$$ with operations $$+$$ and $$ \cdot$$  satisfies both the left and right distributive laws, and so is a ring $$ (B, +, \cdot)$$; and  the circle operation on $$B$$ defined from the multiplication  $$\cdot$$  is the original circle operation, hence $$(B, \circ)$$ is a group, so $$B$$ is a radical ring. Therefore the set of radical rings is a subset of the set of left braces. For some counts of radical rings with n elements for various numbers n, see Radical rings.

Some history A set-theoretic solution of the Yang-Baxter equation (see Yang-Baxter equation) is a pair ($$X$$, $$r$$) where $$X$$ is a set and $$r: X \times X \to X \times X$$ is a bijective map such that . The question of finding set-theoretic solutions of the Yang-Baxter equation was first raised by V. G. Drinfel'd in 1990 [Dr92]. W. Rump [Ru07] defined a left brace as a generalization of a radical ring with the property that a left brace yields a set-theoretic solution of the Y-B equation. Skew left braces (where the “additive” group need not be abelian) were first defined in 2017 by L. Guarneri and L. Vendramin [GV17], who showed that every skew left brace yields a solution of the Y-B equation, and conversely, that every non-degenerate solution of the Yang-Baxter equation corresponds to a unique skew left brace.

Skew left braces and the Yang-Baxter equation

Given a skew brace $$A$$, define $$\sigma_a$$:   $$ A \to A$$ by $$\sigma_a(b) = a^{-1}(a \circ b)$$. Then $$a \circ b = a \sigma_a(b)$$. Define $$R$$ : $$ A$$ × $$A $$ → $$ A$$ × $$A $$ by $$ R(a, b) $$ =  $$(\sigma_a(b), \tau_b(a)) $$ =  $$(\sigma_a(b), \overline{\sigma_a(b)}\circ a \circ b) $$ where $$ \tau_b(a) $$ =  $$ \overline{\sigma_a(b)}\circ a \circ b$$. Then for all $$a$$, $$b$$ in $$A$$, $$\sigma_a$$     and $$\tau_b$$ are one-to-one maps from $$A$$ to $$A$$, and we have:

([GV], Theorem 3.1)    If $$A$$ is a skew left brace, then $$A$$ yields a solution    $$R(a, b)$$ of the Yang-Baxter equation:    for all $$a, b, c$$ in $$A$$, .

Since $$ \sigma_a$$ and $$\tau_b$$ are one-to-one maps from $$A$$ to $$A$$ for all $$a$$, $$b$$ in $$A$$, the solution $$R$$ of the Yang-Baxter equation is called nondegenerate.

Here is a proof of this result, adapted from [LYZ00] to the skew brace setting. There are three equalities to show:

(L:)

(C:)

and

(R:) .

Here is how it goes:

Given a skew brace $$B(\circ, \cdot)$$ ($$\cdot$$ is usually omitted),  the maps $$ \sigma_x(y) = x^{-1}(x \circ y)$$ and $$\tau_y(x) = \overline{\sigma_x(y)} \circ x \circ y$$  satisfy the following three properties (c.f. [GV19]):

i) $$ \sigma $$ is a homomorphism from $$(B, \circ )$$ to $$Perm(A)$$:    for $$x$$, $$y$$, $$z$$ in $$A$$, (this is equivalent to the defining equation for a left brace).

ii) $$\tau $$ is an anti-homomorphism from $$(B, \circ)$$ to $$Perm(A)$$:.

iii) $$\sigma_u(v) \circ \tau_v(u) = u \circ v$$. Thus, since  then if $$R(u, v) = (y, z)$$, then $$u \circ v = y \circ z$$.  These properties suffice to show that $$R$$ satisfies.

The left side of (*) is: . Now

where $$k = \tau_f(d), h = \sigma_d(f), g = \tau_c(e), f = \sigma_e(c), e = \tau_b(a), d = \sigma_a(b)$$ and by property iii).

The right side of (*) is:

,

where $$ w = \tau_r(t), v = \sigma_t(r), t = \tau_q(a), s = \sigma_a(q), r = \tau_c(b), q = \sigma_b(c)$$, and by property iii).

To show that $$h = s$$: , and $$d \circ e = \sigma_a(b) \circ \tau_b(a) = a \circ b$$. So $$h = \sigma_{d \circ e}(c) = \sigma_{a \circ b}(c) = s$$.

To show that $$w = g$$: and . So $$w = \tau_{q \circ r}(a) = \tau_{b \circ c}(a)  = g$$.

To show that $$k = v$$: For any $$u, v$$, if $$R(u, v) =   (x, y)$$, then $$x \circ y =  u \circ v$$. So the left side of equation (*) is $$(h, k, g)$$; the right side is $$(s, v, w)$$, and . Since $$w = g$$, and $$h = s$$ in the group $$(B, o)$$, it follows that $$k = v$$. That completes the proof. For an illustration of what these equalities look like for A a radical algebra, see '''Radical Rings.'''

Skew Braces and Hopf-Galois theory. Generalizing work of [FCC12] on Hopf-Galois structures on Galois extensions of fields whose Galois group is a finite abelian p-group for some prime p, D. Bachiller [Bac16], [Bac18] observed a connection between left braces  $$B= (B, \circ, +)$$ and Hopf-Galois structures of type $$ (B, +)$$ on Galois extensions of fields with Galois group isomorphic to $$ (B, \circ)$$. N. Byott and L. Ventramin [SV18] then showed that every left skew brace $$(B, \circ, \cdot)$$ yields at least one Hopf-Galois structure of type $$(B, \cdot)$$ on a classical Galois extension of fields with Galois group $$N$$ isomorphic to (B, circ), and conversely, given a $$N$$-Galois extension of fields which also has a Hopf Galois structure of type $$N$$, then there exists a skew left brace $$(B, \circ, \cdot)$$  with additive group $$ (B, \cdot)$$ isomorphic to $$N$$ and circle group  $$(B, \circ)$$  isomorphic to $$N$$. A given skew brace with circle group $$N$$ may yield more than one Hopf-Galois structure on a Galois extension with Galois group $$G$$:    the number of Hopf-Galois structures corresponding to a given skew brace relates to the sizes of the automorphism groups of automorphisms of $$G$$ and of $$N$$ and is described in the appendix to [SV18].)

Classification results. A natural question arising in skew brace theory, and independently, in Hopf-Galois theory, is to ask, given a pair $$(G, N)$$ of finite groups, is there a skew brace  $$(B, \circ, \cdot)$$ so that   $$ (B, \circ)  \cong G$$   and   $$(B, \cdot) \cong N $$? In skew brace theory the question was typically posed:    given a skew brace $$B$$ with additive group $$(B, \cdot)$$ isomorphic to $$N$$, what are the possible isomorphism types of groups $$G$$ so that $$(B, \circ)$$ is isomorphic to $$G$$? In Hopf-Galois theory the question was typically posed: given a Galois group $$G$$, what are the possible types $$N$$ of Hopf-Galois structures on a $$G$$-Galois extension? Here are some results on this question. (All groups are finite.) • If $$G$$ is a cyclic group of order p^n where p is an odd prime, then $$N$$ must be isomorphic to $$G$$. [Kohl98] • If $$N$$ is a cyclic group of order p^n where p is an odd prime, then $$G$$ must be isomorphic to $$N$$ [Ru07a]. Neither of these results hold if $$p = 2$$: see [By07] and [Ru07a], [Ru19]. • If $$ p$$ is odd, $$N$$ is an abelian p -group of order p^n and of p -rank m where m+1 < p, and $$N$$ is an abelian p-group, then $$G$$ is isomorphic to $$N$$. [Fe03, FCC12] This was generalized in [Bac16] to yield that if B = (B, \circ, \cdot) is a brace with additive group (B, \cdot) = $$N$$ an abelian group of p -rank m with m+1 < p, then for any b in B , the order of b in (B, \cdot) is equal to the order of b in (B, \circ) = $$N$$. In particular, if $$N$$ is an elementary abelian p -group of order p^m for p an odd prime and m+1< p, then $$G$$ must have exponent p.    •  If $$N$$ is abelian, then $$G$$ is solvable. [ESS99], [Byo13]. • If $$G$$ is a simple group, then $$N$$ must be isomorphic to $$G$$ [By04]. • If $$G$$ = S_n, the symmetric group and n = 5 or > 6 , then $$N$$ must be isomorphic to $$G$$ or to A_n \times C_2 (the direct product of the alternating group A_n and the cyclic group of order 2 [Ts19].    • If $$G$$ is abelian, then $$N$$ must be a metabelian group [By15], [Nas19], [TQ20]    • If $$G$$ is a nilpotent group, then $$N$$ is a solvable group [TQ20].    • If $$G$$ is solvable, then $$N$$ need not be solvable [By15]: there exists a skew brace with circle group isomorphic to $$G = A_4 \times C_5 $$ and additive group $$N = A_5$$, where $$A_n$$is the alternating group (the group of even permutations on $$n$$ symbols) and $$C_n$$ is the cyclic group of order $$n$$.    • If $$G$$ is a nilpotent group of class 2, then there exists a brace with circle group $$G$$: in fact, a radical algebra with circle group $$G$$ [AW73].    • If for some m dividing the order of $$N$$, if $$N$$ has more characteristic subgroups of order $$m$$ than $$G$$ has subgroups of order $$m$$, then there is no skew brace with additive group isomorphic to $$N$$ and circle group isomorphic to $$G$$ [Koh19]. • If $$G$$ is a non-cyclic abelian p-group of order $$p^n$$  with $$n > 2$$ and $$p$$ odd, then there is a skew brace with circle group $$G$$ and non-abelian additive group. [BC12] An open conjecture of N. Byott states that if $$G$$ is insolvable, then $$N$$ cannot be solvable: see [TQ20]. Given groups $$G$$ and N of order $$n$$ for which there is a skew brace with additive group $$N$$ and circle group $$G$$, there are also many results on the number of isomorphism types of skew braces $$B$$with additive and circle groups isomorphic to $$N$$ and $$G$$, resp. In particular, for $$n$$ squarefree, see [AB21].

Bi-skew braces

A set $$B$$ with two group structures $$(B, \circ)$$ and $$(B, \cdot)$$ so that $$B$$ is a skew brace with either group acting as the circle group is called a bi-skew brace. One large set of examples are $$(B, \circ, \cdot)$$  where $$B$$is a nilpotent algebra  $$(B, \cdot, +)$$  of index 3 (that is, for all $$a, b, c$$ in $$B$$, $$a \cdot b \cdot c = 0$$). Another set of examples are $$(B, \circ, \cdot)$$  where $$(B, \cdot)$$ is a semidirect product  $$H \rtimes K$$  for $$H$$, $$K$$ finite subgroups of $$(B, \cdot)$$, and  $$(B, \circ)$$ is the direct product  $$H \times K$$. The concept of bi-skew brace, from [Ch19], has been extended to the concept of brace block, a collection of different group operations on a set $$G$$ so that every ordered pair of operations on $$G$$ makes $$G$$ into a skew brace. See [Koc21] and [CS21] for examples and theory.

See also • Jacobson radical

• Yang-Baxter equation.

Notes For expositions of Hopf-Galois structures and their applications to Galois module theory, see [Ch00] and [CGKKKTU21].

References

[AB21] Alabdali, A., Byott, N. Skew braces of squarefree order, J. Algebra and Its Applications 20, No. 7 (2021).

[AW73] Ault, J., Watters, J.. Circle groups of nilpotent rings, American Math. Monthly 80 (1973), 48-52.

[Bac16]    Bachiller, D., Counterexample to a conjecture about braces,    Journal of Algebra, vol    453 (2016), 160-176.

[Bac18]    Bachiller, D.,    Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks, Journal of Knot Theory and Its Ramifications, Vol. 27 (2018),

[BC12] Byott, N. P., Childs, L. N., Fixed point free pairs of homomorphisms and Hopf-Galois structures, New York J. Math 18 (2012), 707--731.

[By04] Byott, N. P., Hopf-Galois structures on field extensions with simple Galois groups, Bull. London Math. Soc. 36 (2004), 23--29

[By13] Byott, N. P., Nilpotent and abelian Hopf-Galois structures on field extensions, Journal of Algebra 318 (2013), 131-139.

[By15] Byott, N. P.,Solubility criteria for Hopf-Galois structures, New York J. Math 21 (2015), 883-903.

[CS21] Caranti, A., Stefanello, L., From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, and Hopf-Galois structures,  J. Algebra 587 (2021), 462--487.

[Ch00] Childs, L. N.  Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, American Math. Soc. Math. Surveys and Monographs, vol. 80 (2000).

[Ch19] Childs, L. N., Bi-skew braces and Hopf-Galois structures, New York J. Math. 25 (2019), 574--588.

[CGKKKTU21] Childs, L. N., Greither, C., Keating, K. P., Koch, A., Kohl, T., Truman, P. J., Underwood, R. G., Hopf Algebras and Galois Module Theory, American Math. Soc. Math. Surveys and Monographs, vol. 260 (2021).

[Dr92]    Drinfel'd, V., On some unsolved problems in quantum group theory, Lecture Notes in Mathematics 1510 (1992), 1--8.

[ESS99] Etinghof, P., Schedler, T., Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169-209.

[Fe03] Featherstonhaugh, S. C., Hopf algebra structures on abelian Galois extensions of fields, Ph. D. thesis, Univ. at Albany, 2003.

[FCC12]    Featherstonhaugh, S. C., Caranti, A., Childs, L. N., Abelian Hopf Galois structures on prime-power Galois field extensions, Trans. Amer. Math. Soc. 364 (2012), 3675--3684.

[GV17]    Guarnieri, L., Ventramin, L., Skew braces and the Yang-Baxter equation, Math. Comp. 86 (2017), 2519--2534.

[Koc21] Koch, A., Abelian maps, brace blocks, and solutions to the Yang-Baxter equation, arXiv:2102.06104

[Koh98] Kohl, T., Classification of the Hopf Galois structures on prime power radical extensions, J. Algebra 207 (1998), 525-546.

[Koh19] Kohl, T.  Characteristic subgroups lattices and Hopf-Galois structures, Intern. J. Algebra Computation 29 (2019), 391--405.

[LYZ00] Lu, J-H., Yang, M., Zhu, Y.-C., On the set-theoretical Yang-Baxter equation, Duke Math. J. 104 (2000), 1--18.

[Nas19] Nasybullow, L., Connections between properties of the additive and multiplicative groups of a two-sided skew brace, J. Algebra 540 (2019), 156-167.

[Ru07]    Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307 (2007), 153--170.

[Ru07a] Rump, W., Classification of cyclic braces,  J. Pure and Applied Algebra 209 (2007), 671--685.

[Ru19] Rump, W., Classification of cyclic braces II, Trans. Amer. Math. Soc. 372 (2019), 305-328.

[SV18]    Smoktunowicz, A., Vendramin, L., On skew braces (with an appendix by N. Byott and L. Vendramin), J. Combinatorial Algebra 2 (2018), 47--86.

[Tsa19] Tsang, C.,  Hopf-Galois structures on a Galois S_n-extension, J. Algebra (2019), 349--361.

[TQ20] Tsang, C., Qin, C., On the solvability of regular subgroups in the holomorph of a finite solvable group, Internat. J. Algebra Comput. 30 (2020), 253--265.