Racks and Quandles #

This file defines racks and quandles, algebraic structures for sets that bijectively act on themselves with a self-distributivity property. If R is a rack and act : R → (R ≃ R)→ (R ≃ R)≃ R) is the self-action, then the self-distributivity is, equivalently, that

act (act x y) = act x * act y * (act x)⁻¹
⁻¹

where multiplication is composition in R ≃ R≃ R as a group. Quandles are racks such that act x x = x for all x.

One example of a quandle (not yet in mathlib) is the action of a Lie algebra on itself, defined by act x y = Ad (exp x) y.

Quandles and racks were independently developed by multiple mathematicians. David Joyce introduced quandles in his thesis [Joyce1982] to define an algebraic invariant of knot and link complements that is analogous to the fundamental group of the exterior, and he showed that the quandle associated to an oriented knot is invariant up to orientation-reversed mirror image. Racks were used by Fenn and Rourke for framed codimension-2 knots and links in [FennRourke1992]. Unital shelves are discussed in [crans2017].

The name "rack" came from wordplay by Conway and Wraith for the "wrack and ruin" of forgetting everything but the conjugation operation for a group.

Main definitions #

Shelf is a type with a self-distributive action
UnitalShelf is a shelf with a left and right unit
Rack is a shelf whose action for each element is invertible
Quandle is a rack whose action for an element fixes that element
Quandle.conj defines a quandle of a group acting on itself by conjugation.
ShelfHom is homomorphisms of shelves, racks, and quandles.
Rack.EnvelGroup gives the universal group the rack maps to as a conjugation quandle.
Rack.oppositeRack gives the rack with the action replaced by its inverse.

Main statements #

Rack.EnvelGroup is left adjoint to Quandle.Conj (toEnvelGroup.map). The universality statements are toEnvelGroup.univ and toEnvelGroup.univ_uniq.

Implementation notes #

"Unital racks" are uninteresting (see Rack.assoc_iff_id, UnitalShelf.assoc), so we do not define them.

Notation #

The following notation is localized in quandles:

x ◃ y◃ y is Shelf.act x y
x ◃⁻¹ y◃⁻¹ y⁻¹ y is Rack.inv_act x y
S →◃ S'→◃ S'◃ S' is ShelfHom S S'

Use open quandles to use these.

Todo #

If g is the Lie algebra of a Lie group G, then (x ◃ y) = Ad (exp x) x◃ y) = Ad (exp x) x forms a quandle.
If X is a symmetric space, then each point has a corresponding involution that acts on X, forming a quandle.
Alexander quandle with a ◃ b = t * b + (1 - t) * b◃ b = t * b + (1 - t) * b, with a and b elements of a module over Z[t,t⁻¹]⁻¹].
If G is a group, H a subgroup, and z in H, then there is a quandle (G/H;z) defined by yH ◃ xH = yzy⁻¹xH◃ xH = yzy⁻¹xH⁻¹xH. Every homogeneous quandle (i.e., a quandle Q whose automorphism group acts transitively on Q as a set) is isomorphic to such a quandle. There is a generalization to this arbitrary quandles in [Joyce's paper (Theorem 7.2)][Joyce1982].

Tags #

rack, quandle