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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

$$a\circ (b\cdot c)=(a\circ b)\cdot a^{-1}\cdot (a\circ c)$$

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 $bin<math>A$ so that $a\circ b=b\circ a=0$: 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 $A$, $a\circ (b+c)$ = $(a\circ b)-a+(a\circ c)$ becomes, after replacing $x\circ y$ by $x+y+xy$ for all $x,y$ in $A$,

$$a+b+c+a(b+c)$$

 =

$$a+b+ab-a+a+c+ac,$$

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

$$(r\times id)(id\times r)(r\times id)=(id\times r)(r\times id)(id\times r):X\times X\times X\to X\times X\times X$$

. 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 $A$,

$$(R\times id)(id\times R)(R\times id)(a,b,c)=(id\times R)(R\times id)(id\times R)(a,b,c)$$

.

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:) Failed to parse (syntax error): {\displaystyle \sigma_{\sigma_x(y)}(\sigma_{\tau_y(x)}(z))= \sigma_x(\sigma_y(z)), }

(C:) Failed to parse (syntax error): {\displaystyle \tau_{\sigma_{\tau_y(x)}(z)}(\sigma_x(y)) =    \sigma_{\tau_{\sigma_y(z)}(x)}(\tau_z(y)) }

and

(R:) Failed to parse (syntax error): {\displaystyle \tau_z(\tau_y(x))) = \tau_{\tau_z(y)}(\tau_{\sigma_y(z)}(x)) )} .

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 $A$,

$$\sigma _{x\circ y}(z)=\sigma _{x}(\sigma _{y}(z)):(B,\circ )\to Perm(A)$$

; (this is equivalent to the defining equation for a left brace).

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

$$\tau _{x\circ y}=\tau _{y}\tau _{x}:(B,\circ )\to Aut(A)$$

.

iii) $\sigma _{u}(v)\circ \tau _{v}(u)=u\circ v$. Thus, since

$$R(u,v)=(\sigma _{u}(v),\tau _{v}(u))$$

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

$$(R\times id)(id\times R)(R\times id)(a,b,c)=(id\times R)(R\times id)(id\times R)(a,b,c)$$

.

The left side of (*) is:

$$(R\times id)(id\times R)(R\times id)(a,b,c)=(id\times R)(R\times id)(id\times R)(a,b,c)$$

.

Now

$$(R\times id)(id\times R)(R\times id))(a,b,c)=(R\times id)(id\times R)(d,e,c)=(R\times id)(d,f,g)=(h,k,g)$$

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

$$a\circ b=d\circ e,e\circ c=f\circ g,d\circ f=h\circ k$$

by property iii).

The right side of (*) is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (id × R)(R × id)(id × R)(a, b, c) = (id × R)(R × id)(a, q, r) = (id × R)(s, t, r)= (s, v, w)} ,

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

$$b\circ c=q\circ r,a\circ q=s\circ t,t\circ r=v\circ w$$

by property iii).

To show that $h=s$:

$$s=\sigma _{a}(q)=\sigma _{a}(\sigma _{b}(c))=\sigma _{a\circ b}(c)$$

;

$$h=\sigma _{d}(f)=\sigma _{d}(\sigma _{e}(c))=\sigma _{d\circ e}(c)$$

, 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 $w=g$:

$$g=\tau _{c}(e)=\tau _{c}(\tau _{b}(a))=\tau _{b\circ c}(a)$$

;

$$w=\tau _{r}(t)=\tau _{r}(\tau _{q}(a))=\tau _{q\circ r}(a)$$

; and

$$q\circ r=\sigma _{b}(c)\circ \tau _{c}(b)=b\circ c$$

. 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

$$s\circ v\circ w=s\circ t\circ r=a\circ q\circ r=a\circ b\circ c=d\circ e\circ c=d\circ f\circ g=h\circ k\circ g$$

. 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)</math>, 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</math>  where p</math> is an odd prime, then $N$ must be isomorphic to $G$. [Kohl98]
   • If $N$ is a cyclic group of order  p^n</math> where p</math> 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</math>-group of order p^n</math> and of p</math>-rank m</math> where m+1 < p</math>, 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</math> = (B, \circ, \cdot)</math> is a brace with additive group (B, \cdot)</math> = $N$ an abelian group of p</math>-rank m</math> with m+1 < p</math>, then for any b</math> in B</math>, the order of b</math> in (B, \cdot)</math> is equal to the order of b</math> in (B, \circ) = $N$.  In particular, if $N$ is an elementary abelian p</math>-group of order p^m</math> for p</math> an odd prime and m+1< p</math>, then $G$ must have exponent p.</math> 
   •  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</math>, the symmetric group and n = 5</math> or > 6</math>, then $N$ must be isomorphic to $G$</math> or to  A_n \times C_2</math> (the direct product of the alternating group A_n</math> and the cyclic group of order 2 [Ts19].
   • If $G$ is abelian, then $N$ must be a metabelian group [By15], [Nas19], [TQ20]
   • If $G$</math> 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.