This post follows on from this one. The purpose of posts in this category is for me to learn more about the research being done in quantum groups. This post looks at this paper of Schmidt.
Preliminaries
Compact Matrix Quantum Groups
The author gives the definition and gives the definition of a (left, quantum) group action.
Definition 1.2
Let be a compact matrix quantum group and let
be a
. An (left) action of
on
is a unital *-homomorphism
that satisfies the analogue of
, and the Podlés density condition:
.
Quantum Automorphism Groups of Finite Graphs
Schmidt in this earlier paper gives a slightly different presentation of . The definition given here I understand:
Definition 1.3
The quantum automorphism group of a finite graph with adjacency matrix
is given by the universal
-algebra
generated by
such that the rows and columns of
are partitions of unity and:
.
_______________________________________
The difference between this definition and the one given in the subsequent paper is that in the subsequent paper the quantum automorphism group is given as a quotient of by the ideal given by
… ah but this is more or less the definition of universal
-algebras given by generators
and relations
:
where presumably all works out OK, and it can be shown that
is a suitable ideal, a Hopf ideal. I don’t know how it took me so long to figure that out… Presumably the point of quotienting by (a presumably Hopf) ideal is so that the quotient gives a subgroup, in this case
via the surjective *-homomorphism:
.
_______________________________________
Compact Matrix Quantum Groups acting on Graphs
Definition 1.6
Let be a finite graph and
a compact matrix quantum group. An action of
on
is an action of
on
(coaction of
on
) such that the associated magic unitary
, given by:
,
commutes with the adjacency matrix, .
By the universal property, we have via the surjective *-homomorphism:
,
.
Theorem 1.8 (Banica)
Let
, and
,
be an action, and let
be a linear subspace given by a subset
. The matrix
commutes with the projection onto
if and only if
Corollary 1.9
The action
preserves the eigenspaces of
:
Proof: Spectral decomposition yields that each , or rather the projection
onto it, satisfies a polynomial in
:
,
as commutes with powers of
A Criterion for a Graph to have Quantum Symmetry
Definition 2.1
Let . Permutations
are disjoint if
, and vice versa, for all
.
In other words, we don’t have and
permuting any vertex.
Theorem 2.2
Let
be a finite graph. If there exists two non-trivial, disjoint automorphisms
, such that
and
, then we get a surjective *-homomorphism
. In this case, we have the quantum group
, and so
has quantum symmetry.
Proof: Suppose we have disjoing with
and
.
The group algebra of can be given as a universal
-algebra (related to something I am looking at, the relationship between universal
-algebras and simplicity, note that
is simple under suitable assumptions… this is about when a concrete
-algebra is isomorphic to a univeral one, or when two universal
-algebras are isomorphic).
Anyway, we have:
The proof plans to use the universal property (of ) to get a surjective *-homormorphism onto
, and as the
do not commute, this gives quantum symmetries.
Identifying with their permutation matrices, define:
.
Schmidt shows that these satisfies the relations of
and thus we have a *-homormophism.
I thought such maps were automatically surjective… but obviously not. Schmidt shows that it is and so the rest follows
Remark 2.3
Consider so that
and
. The disjoint automorpshisms
gives the famous surjective *-homorphism exhibiting the quantum nature of
. This subgroup is therefore isomorphic to, well its algebra of functions, to
.
Remark 2.4
Any pair of disjoint automorphisms give quantum symmetry,
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