Idea and Intuition
Let be a (usually finite) set of generators and a (usually finite) set of relations between the generators. The generators at this point are indeterminates, and we will be momentarily vague about what is and isn’t a relation. We write (if it exists!) for the universal -algebra generated by generators and relations .
It has the following universal property. Suppose . Let be a -algebra with elements that satisfy the relations , then there is a (unique) *-homomorphism mapping . This map will be a surjective *-homomorphism (aka a quotient map and a quotient of .
If the generate , , then is a surjective *-homomorphism.
My (highly non-rigourous, the tilde reminding of hand-waving) intuition for this object is that you collect all (really all, not just isomorphism classes, we want below and e.g. ) of -algebras generated by generators satisfying the relations and forming a “big direct sum/product thingy” out of all of them:
.
Then the *-homomorphism given by the universal property is given by projection onto that factor (which is a surjective *-homomorphism, a quotient):
.
This intuition works well but we should give a brief account of things are done properly.
First off, not every relation will give a universal -algebra. For example, consider and . The problem here is one of norm. Recall our rough intuition from above. What is the norm on , this big thing ? It should be something like, where the norm of is the supremum over the factors:
The first approach to show that does not exist is to consider for each the -algebra which is singly-generated by the self-adjoint . But the norm in is the supremum norm, so we find . From here:
which is unbounded. The relations must give a bounded norm to the generators.
As an example of relations that do bound the generators, consider, self-adjoint generators such that the sum of their squares is the identity:
.
These relations bounds the norm of the generators 1, and this gives existence to this algebra, using the Gelfand philosophy giving the “algebra of continuous functions on the free sphere”, (I think first considered by Banica and Goswami).
Note here the relations are given by polynomial relations. If the polynomial relations are suitably admissible (i.e. give a bound to the generators), in this setting there is a real construction (real vs our ridiculous ) of . See p.885 (link to *.pdf quantum group lecture notes of Moritz Weber).
In fact, this is only a small class of the possible relations. I suggest there are at least two more types:
- relations that would be (admissible) norm relations (for example, in one generator, adding , a non-polynomial relation, to the polynomial relation gives an admissible set of relations, and . For /norm relations see here and maybe here.
- (admissible) strong relations (see here for a use of this, with reference)
The constructions in one or both of cases might be constructive, as in the case (admissible) polynomial relations, but there is also an approach using category theory. But the main feature in all such definitions is the universal property, whose use could be summarised as follows:
Let be a universal -algebra. The universal property can be used to answer questions about such as:
- is some polynomial in the generators non-zero,
- is infinite dimensional,
- is non-commutative;
because, if is a -algebra with elements that satisfy the relations then there is a unique *-homomorphism . So, for example, where is some such then
- if , then as ,
- if is infinite dimensional then is a surjective *-homomorphism onto an infinite dimensional algebra, and so the domain is infinite dimensional too.
- if the commutator , so that is non-commutative, then so is as
These quotients can be considered models of .
Two Examples
A projection in a -algebra is such that . Consider
.
Existence is easy, because the norm of a non-zero projection is one. To answer questions about this algebra consider the infinite dihedral group . This has group algebra and group -algebra . Note that and in satisfy the relations of , and so we have a *-homomorphism (in fact a *-isomorphism) . This tells us that any monomial in the generators of is non-zero, is infinite dimensional, and is non-commutative.
A partition of unity is a finite set of projections that sum to the identity, . A magic unitary in a -algebra is a matrix such that the entries along any one row or column form a partition of unity. Consider (notation to be kept mysterious):
.
Consider the following magic unitary:
.
Note that the satisfy the relations of and in fact generate from above. Thus we have a quotient which shows that is infinite dimensional and noncommutative.
It is possible to show using a magic unitary with entries in that for , a monomial with entries in is zero for trivial reasons only (link to *.pdf, from Theorem 1 on).
In addition it can be shown that for (and similarly ) the matrix in with – entries
is a magic unitary, and thus by the universal property is a *-homomorphism… the comultiplication giving the structure of a compact quantum group.
Commutative Examples
If a universal algebra is commutative (as in commutativity, , is one of the relations, vs the relations imply commutativity, as in (nice exercise)), we write . In this case Gelfand’s Theorem, that , often allows us to easily identity the algebra (vs the noncommutative case where the universal algebra is mostly studied via models, quotients).
Theorem
If is a (polynomial) universal commutative -algebra, then it of the form , and is given by the tuples that satisfy the relations of of .
Proof: Characters are *-homomorphisms .
Suppose that satisfy the relations. Then by the universal property, is a *-homomorphism.
On the contrary, suppose that is a character. Then the relations are preserved under a *-homomorphism.
Examples
For , projections in are just the scalars zero and one. Thus the spectrum is and is the algebra of continuous functions on four points.
For collect the tuple of generators in a matrix. The relations imply that each such tuple is in fact a permutation matrix, and so the universal algebra above is the algebra of continuous functions on .
For
you end up with tuples of real numbers in whose sum of squares is one… otherwise known as the sphere .
Liberations
An interesting business here is to start with a universal commutative algebra, say one of the three examples above… and see do you get something strictly bigger, necessarily non-commutative, if you drop commutativity. In the above, yes you do. Gelfand’s theorem says that a commutative unital -algebra is the algebra of continuous functions on a compact space (which we call a classical space). The Gelfand Philosophy says therefore that a noncommutative unital -algebra can be thought of as the algebra of continuous functions on a compact quantum space . Note here is not a set-of-points, but a virtual object, and strictly is just notation (but see here).
Liberating the second example above from commutativity is the passage from the permutation group to the quantum permutation group . Liberating the third example above gives the passage from the real sphere to a quantum sphere called the free sphere .
We can also, of course, work in the other direction, imposing commutativity on not-necessarily universal . If we write , then imposing commutativity gives us the classical version of .
Imposing commutativity is not so scary: using the above you just get … and everything we said above about identifying characters on holds for too.
This can be used: for example if a quantum group acts on a structure , then its classical version acts on . This idea was used by Banica and I to show that not every quantum permutation group is the quantum automorphism group of a finite graph (link to *.pdf).
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