Affine and quasi-affine quandles
Přemysl Jedlička1 , Agata Pilitowska2
David Stanovský3 , Anna Zamojska-Dzienio2
1 Department
of Mathematics, Faculty of Engineering, Czech University of Life
Sciences
2 Faculty of Mathematics and Information Science, Warsaw University of
Technology
3 Department of Algebra, Faculty of Mathematics and Physics, Charles University
Groups, Rings and the Yang-Baxter equation
Spa, 2017
Quasi-affine quandles
Quandles
A binary algebra (Q, ·) is called a quandle if it is:
a left quasigroup: the equation x · u = y has a unique
solution u ∈ Q for every x, y ∈ Q
left distributive: x · (y · z) = (x · y ) · (x · z) for every
x, y , z ∈ Q
idempotent: x · x = x for each x ∈ Q
Spa, 2017
Quasi-affine quandles
Quandles
A binary algebra (Q, ·) is called a quandle if it is:
a left quasigroup: the equation x · u = y has a unique
solution u ∈ Q for every x, y ∈ Q
left distributive: x · (y · z) = (x · y ) · (x · z) for every
x, y , z ∈ Q
idempotent: x · x = x for each x ∈ Q
All left translations La : Q → Q; La (x) = a · x are
automorphisms of Q.
Spa, 2017
Quasi-affine quandles
Quandles
A binary algebra (Q, ·) is called a quandle if it is:
a left quasigroup: the equation x · u = y has a unique
solution u ∈ Q for every x, y ∈ Q
left distributive: x · (y · z) = (x · y ) · (x · z) for every
x, y , z ∈ Q
idempotent: x · x = x for each x ∈ Q
All left translations La : Q → Q; La (x) = a · x are
automorphisms of Q.
Example
(G , ·) - a group,
x . y := x · y · x −1 ,
Conj G := (G , .) is a conjugation quandle.
Spa, 2017
Quasi-affine quandles
Affine quandles
(A, +) - an abelian group, f ∈ Aut(A). A quandle
Aff(A, f ) = (A, ∗), where
x ∗ y = x − f (x) + f (y ) = (1 − f )(x) + f (y )
is called an affine quandle (or Alexander quandle).
Spa, 2017
Quasi-affine quandles
Affine quandles
(A, +) - an abelian group, f ∈ Aut(A). A quandle
Aff(A, f ) = (A, ∗), where
x ∗ y = x − f (x) + f (y ) = (1 − f )(x) + f (y )
is called an affine quandle (or Alexander quandle).
Quandles embeddable into affine quandles are called quasi-affine
ones.
Spa, 2017
Quasi-affine quandles
Affine quandles
(A, +) - an abelian group, f ∈ Aut(A). A quandle
Aff(A, f ) = (A, ∗), where
x ∗ y = x − f (x) + f (y ) = (1 − f )(x) + f (y )
is called an affine quandle (or Alexander quandle).
Quandles embeddable into affine quandles are called quasi-affine
ones.
How to recognize them?
Spa, 2017
Quasi-affine quandles
The group of displacements
Let Q be a quandle. The group of displacements is the group
Dis(Q) = hLa L−1
b | a, b ∈ Qi.
Spa, 2017
Quasi-affine quandles
The group of displacements
Let Q be a quandle. The group of displacements is the group
Dis(Q) = hLa L−1
b | a, b ∈ Qi.
The orbits of the natural action of Dis(Q) on Q, are called orbits
of the quandle Q:
Qe = {α(e) : α ∈ Dis(Q)}.
Orbits are subquandles of Q. They form a block system - the orbit
decomposition of Q.
Spa, 2017
Quasi-affine quandles
The group of displacements
Let Q be a quandle. The group of displacements is the group
Dis(Q) = hLa L−1
b | a, b ∈ Qi.
The orbits of the natural action of Dis(Q) on Q, are called orbits
of the quandle Q:
Qe = {α(e) : α ∈ Dis(Q)}.
Orbits are subquandles of Q. They form a block system - the orbit
decomposition of Q.
A permutation group G acting on a set X is semiregular if
non-trivial permutations from G have no fixed points.
Spa, 2017
Quasi-affine quandles
Characterization theorem I
Theorem
A quandle Q is quasi-affine iff
Dis(Q) is abelian and semiregular.
Spa, 2017
Quasi-affine quandles
Characterization theorem I
Theorem
A quandle Q is quasi-affine iff
Dis(Q) is abelian and semiregular.
For finite quandles: a quasi-affine quandle is affine iff, in every
column in the multiplication table, each entry has the same
number of occurences.
Spa, 2017
Quasi-affine quandles
Quasi-affine quandle which is not affine.
Example
(Q, ·)
·
0
1
2
3
4
5
0
0
0
0
1
1
1
1
1
1
1
2
2
2
2
2
2
2
0
0
0
3
4
4
4
3
3
3
4
5
5
5
4
4
4
5
3
3
3
5
5
5
Q is not an affine quandle
Dis(Q) = h(0 1 2)(3 5 4)i
Q is quasi-affine
Spa, 2017
Quasi-affine quandles
Semiregular extensions
Definition
(A, +) - an abelian group, f ∈ Aut(A), I - a non-empty set, di ∈ A
for i ∈ I . The quandle Ext(A, f , (di : i ∈ I )) = (I × A, ∗), where
(i, a) ∗ (j, b) = (j, (1 − f )(a) + f (b) + di − dj ),
is a semiregular extension over Aff(A, f ).
Spa, 2017
Quasi-affine quandles
Semiregular extensions
Definition
(A, +) - an abelian group, f ∈ Aut(A), I - a non-empty set, di ∈ A
for i ∈ I . The quandle Ext(A, f , (di : i ∈ I )) = (I × A, ∗), where
(i, a) ∗ (j, b) = (j, (1 − f )(a) + f (b) + di − dj ),
is a semiregular extension over Aff(A, f ).
The extension is indecomposable, if the fibers {i} × A are the
orbits of Ext(A, f , (di : i ∈ I )).
Spa, 2017
Quasi-affine quandles
Semiregular extensions for quasi-affine quandles
Every quasi-affine quandle Q admits a representation as an
indecomposable semiregular extension.
Q∼
= Ext(Dis(Q), f , (Lx L−1
e : x ∈ T )),
where e ∈ Q, T is a transversal of the orbit decomposition of Q,
and f is the automorphism of Dis(Q) defined by
f (α) = αLe = Le αL−1
e .
Spa, 2017
Quasi-affine quandles
Characterization theorem II
Theorem
A quandle Q is quasi-affine iff
¯ for some abelian group A, its
Q is isomorphic to Ext(A, f , d)
automorphism f and some tuple d¯ = (di : i ∈ I ) of elements of A.
Spa, 2017
Quasi-affine quandles
Characterization theorem II
Theorem
A quandle Q is quasi-affine iff
¯ for some abelian group A, its
Q is isomorphic to Ext(A, f , d)
automorphism f and some tuple d¯ = (di : i ∈ I ) of elements of A.
Theorem
A quandle Q is affine iff
¯ for some abelian group A, its
Q is isomorphic to Ext(A, f , d)
automorphism f and some tuple d¯ = (di : i ∈ I ) of elements of A
which forms a multitransversal of A/Im(1 − f ).
Spa, 2017
Quasi-affine quandles
Quasi-affine quandle which is not affine.
Example
(Q, ·)
·
0
1
2
3
4
5
0
0
0
0
1
1
1
1
1
1
1
2
2
2
2
2
2
2
0
0
0
3
4
4
4
3
3
3
4
5
5
5
4
4
4
5
3
3
3
5
5
5
Q∼
= Ext(Z3 , 1, (0, 1)) is not an affine quandle: Im(1 − f ) = {0}
¯
and {2} has no representative in d.
Spa, 2017
Quasi-affine quandles
Isomorphism theorem
Theorem
Let Ext(A, f , (di : i ∈ I )) and Ext(A0 , f 0 , (di0 : i ∈ I 0 )) be two
indecomposable extensions. They are isomorphic if and only if
there exist a bijection π : I → I 0 , an isomorphism ψ : A → A0 , and
an element a ∈ A0 such that
(E1) ψf = f 0 ψ,
0
(E2) ψ(di ) − dπ(i)
∈ a + Im(1 − f 0 ) for every i ∈ I .
Spa, 2017
Quasi-affine quandles
Enumeration
Example
There are four quasi-affine quandles of order six:
Aff(Z6 , 1) ' Ext(Z1 , 1, (0, 0, 0, 0, 0, 0)),
Ext(Z2 , 1, (0, 0, 1)),
Ext(Z3 , 1, (0, 1)),
Aff(Z6 , −1) ' Ext(Z3 , 2, (0, 0)).
Spa, 2017
Quasi-affine quandles
Enumeration
Example
There are four quasi-affine quandles of order six:
Aff(Z6 , 1) ' Ext(Z1 , 1, (0, 0, 0, 0, 0, 0)),
Ext(Z2 , 1, (0, 0, 1)),
Ext(Z3 , 1, (0, 1)),
Aff(Z6 , −1) ' Ext(Z3 , 2, (0, 0)).
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
quasi-affine 1 1 2 3 4 4 6 9 12 7 10 17 12 10 14
affine 1 1 2 3 4 2 6 7 11 4 10 6 12 6 8
Figure: The number of quasi-affine and affine quandles of order n, up to
isomorphism.
Spa, 2017
Quasi-affine quandles