Introduction
Simple left braces
Matched products of left braces and simplicity
Author: Ferran Cedó
Universitat Autònoma de Barcelona
(Joint work with David Bachiller, Eric Jespers and Jan Okniński)
Spa, June 2017
Introduction
Simple left braces
Solutions of the YBE
Definition
Let X be a non-empty set. A set-theoretic solution of the
Yang-Baxter equation on X is a bijective map r : X × X → X × X
such that
r1 r2 r1 = r2 r1 r2 ,
where r1 = r × idX and r2 = idX ×r are maps from X × X × X to
it self.
We write r (x, y ) = (σx (y ), γy (x)), for all x, y ∈ X .
The map r is involutive if r 2 = idX 2 .
We say that r is non-degenerate if the maps σx , γx : X → X are
bijective, for all x ∈ X .
Convention. By a solution of the YBE we mean an involutive
non-degenerate set-theoretic solution of the Yang-Baxter equation.
Introduction
Simple left braces
Braces and the Yang-Baxter equation
In 2007 Rump introduced braces as a generalization of radical rings
to study solutions of the YBE. In 2016, the classification of such
solutions has been reduced to the classification of left braces
(Bachiller, C., Jespers). The following definition is equivalent to
the original definition of Rump.
Definition
A left brace is a set B with two binary operations, + and ·, such
that (B, +) is an abelian group, (B, ·) is a group, and for every
a, b, c ∈ B,
a · (b + c) + a = a · b + a · c.
Note that in a left brace B, 1 = 0 (taking a = 1 and b = c = 0 in
the above formula).
In any left brace B there is an action λ : (B, ·) → Aut(B, +)
defined by λ(a) = λa and λa (b) = ab − a, for a, b ∈ B.
Introduction
Simple left braces
Braces
Definition
An ideal of a left brace B is a normal subgroup I of the
multiplicative group of B such that λa (b) ∈ I for all b ∈ I and all
a ∈ B.
It is easy to see that every ideal I of a left brace B also is a
subgroup of the additive group of B, and then B/I is a left brace,
the quotient brace B modulo I .
Definition
A non-zero left brace B is simple if {0} and B are the only ideals
of B.
Theorem (Etingof, Schedler, Soloviev)
The multiplicative group of a finite left brace is solvable.
Introduction
Simple left braces
Previously known finite simple left braces
- Rump (2007): Every simple left brace of prime power order p n
is a trivial brace of cardinality p.
- Bachiller (2015): He constructed the first finite nontrivial
simple left braces with additive groups isomorphic to
Z/(p1 ) × (Z/(p2 ))k(p1 −1)+1 ,
where p1 , p2 are primes such that p2 | p1 − 1 and k is a
positive integer.
He introduced the concept of matched product of two left
braces and noted that every finite left brace of order mn, with
m and n coprime, is a matched product of two left braces of
orders m and n.
Introduction
Simple left braces
Matched product of left braces
Definition
A left ideal of a left brace B is a subgroup I of its additive group
such that λa (b) ∈ I , for all a ∈ B and all b ∈ I .
It is easy to see that every left ideal I of a left brace B also is a
subgroup of the multiplicative group of B.
Definition
We say that a left brace B is a matched product of left ideals
I1 , I2 , . . . , In if the additive group of B is the direct sum of these
left ideals.
Note that if B is a finite left brace, then every Sylow subgroup of
its additive group is a left ideal and thus B is the matched product
of these left ideals.
Introduction
Simple left braces
Matched product of left braces
Theorem
Let B1 , . . . , Bn be left braces with n ≥ 2. Let
α(j,i ) : (Bj , ·) −→ Aut(Bi , +)
be actions satisfying the following conditions.
(1)
(j,k) (i ,k)
αb
α
(j,i )
(a)
b −1
α
(i ,k) (j,k)
= αa
α
(i ,j)
(b)
a−1
α
,
for all a ∈ Bi and b ∈ Bj , and for all i , j, k ∈ {1, . . . , n},
(i ,j)
where α(i ,j) (a) = αa .
(2) For every i ∈ {1, . . . , n}, α(i ,i ) is the lambda map of Bi , that
(i ,i )
is, αx (y ) = xy − x, for all x, y ∈ Bi .
Introduction
Simple left braces
Matched product of left braces
Then there exists a unique structure of left brace on B1 × · · · × Bn
such that
(i) (a1 , . . . , an ) + (b1 , . . . , bn ) = (a1 + b1 , . . . , an + bn ), for all
ai , bi ∈ Bi and i = 1, . . . , n.
(ii) λ(0,...,0,ai ,0,...,0) (0, . . . , 0, bj , 0, . . . , 0) =
(i ,j)
(0, . . . , 0, αai (bj ), 0, . . . , 0), for all ai ∈ Bi , bj ∈ Bj and all
i , j ∈ {1, . . . , n}.
Furthermore, each {0} × · · · × {0} × Bi × {0} × · · · × {0} is a left
ideal of this left brace.
We denote this left brace by B1 ⊲⊳ . . . ⊲⊳ Bn and we say that it is
the matched product of the left braces B1 , . . . , Bn via the actions
α(i ,j) .
Introduction
Simple left braces
Hegedűs left braces
Let p be a prime. Hegedűs constructed a left brace H(p, n, Q, f )
with additive group (Z/(p))n+1 (n is a positive integer) and with
the lambda map defined by
λ(~x ,µ) (~y , µ′ ) = (f q(~x ,µ) (~y ), µ′ + b(~x , f q(~x ,µ) (~y ))),
for all ~x , ~y ∈ (Z/(p))n and µ, µ′ ∈ Z/(p), where
- Q is a non-degenerate quadratic form over (Z/(p))n ,
- f is an element of order p in the orthogonal group determined
by Q,
- q(~x , µ) = µ − Q(~x ) (with µ ∈ Z/(p)),
- b(~x , ~y ) = Q(~x + ~y ) − Q(~x ) − Q(~y ).
Introduction
Simple left braces
New finite simple left braces
Let s be an integer greater than 1 and let p1 , p2 , . . . , ps be
different odd prime numbers. For 1 ≤ i ≤ s, let
Hi = Hi (pi , ni , Qi , fi ) be the left braces of Hegedűs.
For 1 ≤ i < s, suppose ci is an element of order pi +1 in the
orthogonal group determined by Qi , cs is an element of order p1 in
the orthogonal group determined by Qs such that fi ci = ci fi , for
1 ≤ i ≤ s. For 1 ≤ i , j ≤ s, and i 6= j, define the maps
(i ,j)
α(i ,j) : (Hi , ·) −→ Aut(Hj , +) : (~xi , µi ) 7→ α(~xi ,µi ) ,
with
(k+1,k)
q
α(~xk+1 ,µk+1 ) (~xk , µk ) = (ck k+1
(1,s)
q (~x1 ,µ1 )
α(~x1 ,µ1 ) (~xs , µs ) = (cs 1
(i ,j)
and α(~xi ,µi ) = idHj otherwise.
(~xk+1 ,µk+1 )
(~xs ), µs ),
(~xk ), µk ), for 1 ≤ k < s,
Introduction
Simple left braces
New finite simple left braces
Theorem (Bachiller, C., Jespers, Okniński)
With the above notation, if α(i ,i ) : (Hi , ·) −→ Aut(Hi , +) denotes
the lambda map of Hi , then
(j,k) (i ,k)
αb
α
(j,i )
(a)
b −1
α
(i ,k) (j,k)
= αa
α
(i ,j)
(b)
a−1
α
,
for all a ∈ Hi and b ∈ Hj , and for all i , j, k ∈ {1, . . . , s}. Therefore
the maps α(i ,j) , with 1 ≤ i , j ≤ s, define a matched product of left
braces H1 ⊲⊳ . . . ⊲⊳ Hs .
Introduction
Simple left braces
New finite simple left braces
Theorem (Bachiller, C., Jespers, Okniński)
With the above notation, the matched product of left braces
H1 ⊲⊳ . . . ⊲⊳ Hs via the maps α(i ,j) is simple if and only if ci − id is
an automorphism for all 1 ≤ i ≤ s.
Theorem (Bachiller, C., Jespers, Okniński)
Let s be an integer greater than 1 and let p1 , p2 , . . . , ps be
different odd prime numbers. Then there exists a simple left brace
with additive group
(Z/(p1 ))p1 (p2 −1)+1 ×· · ·×(Z/(ps−1 ))ps−1 (ps −1)+1 ×(Z/(ps ))ps (p1 −1)+1 .
Introduction
Simple left braces
Open problems
Problem
Describe the structure of all left braces of order p n , for a prime p.
And describe the group Aut(B, +, ·) of automorphisms for all such
left braces.
Problem
Determine for which prime numbers p, q and positive integers
α, β, there exists a simple left brace of cardinality p α q β .
References.
- D. Bachiller, F. Cedó, E. Jespers and J. Okniński, Iterated
matched products of finite braces and simplicity; new
solutions of the Yang-Baxter equation, Trans. Amer. Math.
Soc, to appear, arXiv:1610.00477v1[math.GR].
Introduction
Simple left braces
Thank you for your attention!