Simply connected Latin quandles
Marco Bonatto (Charles University, Prague), Petr Vojtěchovský (DU, Denver)
[email protected]
[email protected]
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1
Proposition 2. Let X be a quandle, S a set and β, β 0 ∈ Zc2(X, S). Then TFAE:
Introduction
6
Simply connected latin quandles
(i) β ∼ β 0,
Quandles are groupoids introduced by Joyce [4] as algebraic objects that can be used to
obtain invariants of oriented knots ([5]).
We are interested in extension theory of quandles, with a particular focus on a special class
of extensions of connected quandles. A connected quandle Y is a covering of a quandle
X if X is a factor of Y by a congruence α of Y such that refines the congruence ker(LY ),
where (x, y) ∈ ker(LY ) if and only if the left translations Lx, Ly on Y coincide. It turns out
that the structure of a covering of X is described by the so-called (constant) quandle cocycle.
Connected quandles which admits just the trivial cocycle are called simply connected.
We develop a combinatorial approach to latin quandle cocycles in order to show the following result:
Theorem 1. Let X be a finite connected quandle that is affine over a cyclic group, or a finite
doubly transitive quandle of size different from 4. Then X is simply connected.
i.e. quandles belonging to such classes have just trivial coverings (extending a result given
in [3] for quandles of prime size). Moreover, using a categorical characterization given in [2]
we give an alternative characterization of finite simply connected quandles.
(ii) there exists an isomorphisms φ : X ×β S −→ X ×β 0 S such that the following diagram is
commutative
/
X ×β S π
7X
π
φ
X ×β 0 S
Example 2. Let X be a quandle. The map 1 : (x, y) 7→ 1 is a cocycle and X ×1 S = X × PS is
called trivial covering.
6.1
4
Theorem 4. Let X = Q(Znq, α) be a doubly transitive quandle with |X| 6= 4. Then X is simply
connected.
Characterization of simply connected quandles
Definition 5. A connected quandle X is simply connected if every covering of X is trivial up
to isomorphism, that is, if Hc2(X, S) = {[1]∼} for every non-empty set S, i.e. X admits only
trivial coverings.
Proposition 3. [2] The assignment Adj : Qnd → Grps,
R = {xyx−1 = x · y | x, y ∈ X} is the left adjoint of Conj.
2
Quandles
Definition 1. A groupoid (X, ·) is a quandle if the map Lx : y 7→ x · y is a bijection of X for
every x ∈ X and
x · (y · z) ≈ (x · y) · (x · z),
Example 1. (i) Let S be a set. Then PS = (S, ·), where x · y = y, is called projection quandle.
(ii) Let G be a group and H a subset of G closed under conjugation. Then Conj(H) = (H, ·),
where x · y = xyx−1, is called conjugation quandle on H.
Moreover Conj : Grps → Qnd, G 7→ Conj(G) is a functor.
X 7→ Adj(X) = hX | Ri, where
Lemma 2. [2] Let X be a quandle. Then Adj(X) ' Adj(X)0 oZ, where Adj(X)0 = {xa1 1 . . . xann |
Pn
bX such that
a
=
0}.
Moreover
there
exists
L
i
i=1
X
x · x ≈ x.
A quandle X is connected if LMlt(X) = hLx, x ∈ Xi acts transitively on X, and doubly
transitive if LMlt(X) acts doubly transitively on X. The group Dis(X) = hLxL−1
y , x, y ∈ Xi is
called displacement group of X.
For a latin quandle X, X × X splits into g-orbits Og on which hf, hi since g ∈ Z(G). By Lemma
f
4, the set Og splits into f -invariant subsets Og (u, u) Og0 , Og and their complement in Og . If
X = Q(A, α) is a principal (affine) quandle, the size of the f and h orbits can be computed, in
terms of the size of cycles of α and the order of the elements of A.
i / Adj(X)
LX
bX
L
'
Doubly transitive quandles
Theorem 3. [6] Let X be a finite quandle. Then LMlt(X) is doubly transitive if and only if
X∼
= Q(Znq, α) for some prime q, some n > 0 and α ∈ Aut(Znq) with |α| = |X| − 1.
Proof. In the doubly transitive case, there are at most five orbits of hf, hi, namely {Og (u, u)},
f
Og0 , Og and certain sets Og1 , Og2 . Using the cocycle condition and the action of h, β = 1.
Proposition 5. Let X = Q(Z22, α) be a doubly transitive quandle of size 4 and let 1 = (0, 0), 2 =
(1, 0), 3 = (1, 1), 4 = (0, 1). Then H 2(X, S) = {[βσ ]∼ | σ ∈ Sym(S), σ 2 = 1}, where


1 1 1 1
1 1 σ σ

βσ = 
(1)
1 σ 1 σ
1 σ σ 1
and βσ ∼ βτ if and only if σ and τ are conjugate.
LMlt(X)
0 −→ Dis(X) is a central group extension.
bX |
and ker(L
)
−→
Adj(X)
0
Adj(X)
Theorem 2. Let X be a finite connected quandle. The following are equivalent:
(i) X is simply connected;
6.2
Affine quandles over cyclic groups
Graña showed:
Proposition 6. [3] Let q be a prime and X = Q(Zq , α) an affine quandle. Then X is simply
connected.
(ii) Adj(X)0 is regular;
(iii) X is principal and Dis(X) ' Adj(X)0 .
Lemma 5. Let X = Q(A, α) be a latin affine quandle and let β be a 0-normalized cocycle.
Then h(x, y) = (y + x, y) and β(nx, x) = 1 for every integer n and every x, y ∈ X.
Proposition 7. Let X = Q(Zm, λn) be a latin affine quandle and let β be a 0-normalized
cocycle. Then m is odd and β is u-normalized for every u ∈ U(Zm).
(iii) Let G be a group, α ∈ Aut(G). Then Q(G, α) = (G, ·) where x · y = xα(x−1y), is called
principal quandle. If G is Abelian Q(G, α) is called affine.
5
(iv) Let X be a quandle and let L(X) = {Lx | x ∈ X}. Then Conj(L(X)) is a quandle and the
map LX : X −→ L(X), x 7→ Lx is a quandle morphism.
Definition 6. A quandle is latin if Rx : y 7→ y · x is bijective for every x ∈ X. Let u ∈ X. Then
β ∈ Zc2(X, S) is said to be u-normalized if β(x, u) = 1 for every x ∈ X.
Lemma 6. Let (X, ·) = Q(Zm, α) be a latin affine quandle with m odd. Then for every x ∈ X
there are u, v ∈ U(Zm) such that x = u · v.
Proposition 4. Let X be a latin quandle, S a non-empty set, u ∈ X and β ∈ Zc2(X, S). Then
there exists a u-normalized cocycles βu cohomologous to β and β ∼ 1 if and only if βu = 1.
Theorem 5. Let X = Q(Zm, λn) be a latin affine quandle. Then X is simply connected.
3
Quandle coverings
Normalized Cocycles for latin quandles
Lemma 3. Let X be a latin quandle and let β be a u-normalized cocycle. Define:
Definition 2. A connected quandle Y is a covering of a quandle X if X ∼
= Y /α, where α is a
congruence of Y that refines ker(LY ).
Definition 3. [1] Let X be a quandle and S be a non-empty set. A (constant) quandle cocycle
β ∈ Zc2(X, S) is a map β : X × X → Sym(S) such that
β(xy, xz)β(x, z) = β(x, yz)β(y, z),
β(x, x) = 1
f : X × X → X × X,
g : X × X → X × X,
h : X × X → X × X,
(x, y) 7→ (x · y/u, xu),
(x, y) 7→ (ux, ux),
(x, y) 7→ (y/(x\u) · x, y).
and let G = hf, g, hi. Then β(u, x) = 1 and β(x, y) = β(k(x, y)) for every x, y ∈ X and every
k ∈ G.
The set of all quandle cocycles X × X → Sym(S) will be denoted by Zc2(X, S).
Lemma 1. Let X be a quandle, S a non-empty set and β ∈ Zc2(X, S). Then (X × S, ·), where
(x, s) · (y, t) = (xy, β(x, y)(t)) is a quandle and the projection onto X is a morphism.
Proposition 1. [1] Let X, Y be connected quandles. Then Y is a covering of Xif and only if
Y ∼
= X ×β S for some set S and β ∈ Zc2(X, S).
Remark 1. Note that f (x, y) = (x, y) if and only if y = xu and that f preserves the product
p(x, y) = x · y, i.e., pf (x, y) = p(x, y). On the other hand p(h(x, y)) = p(x, y) if and only if
h(x, y) = (x, y) which is equivalent to x = u.
Moreover g ∈ Z(G), so that hf, hi acts on the set Og = {Og (x, y) | x, y ∈ X} of g-orbits.
Definition 4. Let X be a quandle and S a non-empty set. We say that β, β 0 ∈ Z 2(X, S) are
cohomologous, and we write β ∼ β 0, if there exists γ : X → Sym(S) such that
u 6= x ∈ X}. Then:
(i) h(Ogu) ∩ Ogu = ∅,
β 0(x, y) = γ(xy)β(x, y)γ(y)−1
(ii) h(Og ) ∩ Og = ∅,
holds for every x, y ∈ X and s ∈ S. The set Hc2(X, S) = Zc2(X, S)/ ∼ is the second non-abelian
cohomology of X.
f
Lemma 4. Let X be a latin quandle and Og = {Og (x, xu) | u 6= x ∈ X}, Ogu = {Og (x, x\u) |
f
References
[1] Andruskiewitsch N., Graña M., From racks to pointed Hopf algebras, Advances in Mathematics 178 (2), 177–243 (2003).
[2] Eisermann M., Quandle coverings and their Galois correspondence, Fundamenta Mathematicae 225 (2014), no. 1, 103–168.
[3] Graña M., Indecomposable racks of order p2, Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry 45 (2004), no. 2, 665–676.
[4] Joyce D., A Classifying invatiant of knots, the knot quandle, Journal of Pure and Applied
Algebra 23 (1982) 37–65, North-Holland Publishing Company.
[5] Matveev S. V., Distributive groupoids in knot theory, Math. USSR - Sbornik 47/1 (1984),
73–83.
f
f
f
(iii) f (Og (x, xu)) = Og (x, xu) for every x ∈ X, in particular, f (Og ) = Og ,
(iv) f (Ogu) = Ogu.
Proof. Let β be a 0-normalized cocycle. By Proposition 7, β(x, y) = β(u·v, u·u\y) = β(v, u\y) =
1.
[6] Vendramin L., Doubly Transitive Group and Quandles, accepted for publication in J. Math.
Soc. Japan.