14c. Growth

[math] \newcommand{\mathds}{\mathbb}[/math]

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As an application of all this, corepresentation theory used for “discrete” questions, we can introduce the notion of growth for the discrete quantum groups, as follows:

Definition

Given a closed subgroup [math]G\subset U_N^+[/math], with [math]1\in u=\bar{u}[/math], consider the series whose coefficients are the ball volumes on the corresponding Cayley graph,

[[math]] f(z)=\sum_kb_kz^k\quad,\quad b_k=\sum_{l(v)\leq k}\dim(v)^2 [[/math]]

and call it growth series of the discrete quantum group [math]\widehat{G}[/math]. In the group dual case, [math]G=\widehat{\Gamma}[/math], we obtain in this way the usual growth series of [math]\Gamma[/math].

There are many things that can be said about the growth, and we will be back to this in a moment, with explicit examples, and some general theory as well.

As a first result here, in relation with the notion of amenability, we have:

Theorem

Polynomial growth implies amenability.

Show Proof

We recall from Theorem 14.11 above that the Cayley graph of [math]\widehat{G}[/math] has by definition the elements of [math]Irr(G)[/math] as vertices, and the distance is as follows:

[[math]] d(v,w)=\min\left\{k\in\mathbb N\Big|1\subset\bar{v}\otimes w\otimes u^{\otimes k}\right\} [[/math]]

By taking [math]w=1[/math] and by using Frobenius reciprocity, the lenghts are given by:

[[math]] l(v)=\min\left\{k\in\mathbb N\Big|v\subset u^{\otimes k}\right\} [[/math]]

By Peter-Weyl we have a decomposition as follows, where [math]B_k[/math] is the ball of radius [math]k[/math], and [math]m_k(v)\in\mathbb N[/math] are certain multiplicities:

[[math]] u^{\otimes k}=\sum_{v\in B_k}m_k(v)\cdot v [[/math]]

By using now Cauchy-Schwarz, we obtain the following inequality:

[[math]] \begin{eqnarray*} m_{2k}(1)b_k &=&\sum_{v\in B_k}m_k(v)^2\sum_{v\in B_k}\dim(v)^2\\ &\geq&\left(\sum_{v\in B_k}m_k(v)\dim(v)\right)^2\\ &=&N^{2k} \end{eqnarray*} [[/math]]

But shows that if [math]b_k[/math] has polynomial growth, then the following happens:

[[math]] \limsup_{k\to\infty}\, m_{2k}(1)^{1/2k}\geq N [[/math]]

Thus, the Kesten type criterion applies, and gives the result.

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Let us discuss now as well, as a continuation of all this, the notions of connectedness for [math]G[/math], and no torsion for [math]\widehat{\Gamma}[/math]. These two notions are in fact related, as follows:

Theorem

For a closed subgroup [math]G\subset U_N^+[/math] the following conditions are equivalent, and if they are satisfied, we call [math]G[/math] connected:

• There is no finite quantum group quotient, as follows:
  [[math]] G\to F\neq\{1\} [[/math]]
• The following algebra is infinite dimensional, for any corepresentation [math]v\neq1[/math]:
  [[math]] A_v= \lt v_{ij} \gt [[/math]]

In the classical case, [math]G\subset U_N[/math], we recover in this way the usual notion of connectedness. For the group duals, [math]G=\widehat{\Gamma}[/math], this is the same as asking for [math]\Gamma[/math] to have no torsion.

Show Proof

The above equivalence comes from the fact that a quotient [math]G\to F[/math] must correspond to an embedding [math]C(F)\subset C(G)[/math], which must be of the form:

[[math]] C(F)= \lt v_{ij} \gt [[/math]]

Regarding now the last two assertions, the situation here is as follows:

(1) In the classical case, [math]G\subset U_N[/math], it is well-known that [math]F=G/G_1[/math] is a finite group, where [math]G_1[/math] is the connected component of the identity [math]1\in G[/math], and this gives the result.

(2) As for the group dual case, [math]G=\widehat{\Gamma}[/math], here the irreducible corepresentations are 1-dimensional, corresponding to the group elements [math]g\in\Gamma[/math], and this gives the result.

■

Along the same lines, and at a more specialized level, we can talk as well about the connected component of the identity [math]G_0\subset G[/math], obtained at the algebra level by dividing the Woronowicz algebra [math]C(G)[/math] by a suitable Hopf ideal, as to make dissapear the corepresentations [math]v[/math] such that [math]A_v[/math] is finite dimensional. See Pinzari et al. , .

Finally, once again in connection with all the above, we can talk as well about normal subgroups, and about simple compact quantum groups, as follows:

Definition

Given a quantum subgroup [math]H\subset G[/math], coming from a quotient map [math]\pi:C(G)\to C(H)[/math], the following are equivalent:

• The following algebra satisfies [math]\Delta(A)\subset A\otimes A[/math]:
  [[math]] A=\left\{a\in C(G)\Big|(id\otimes\pi)\Delta(a)=a\otimes1\right\} [[/math]]
• The following algebra satisfies [math]\Delta(B)\subset B\otimes B[/math]:
  [[math]] B=\left\{a\in C(G)\Big|(\pi\otimes id)\Delta(a)=1\otimes a\right\} [[/math]]
• We have [math]A=B[/math], as subalgebras of [math]C(G)[/math].

If these conditions are satisfied, we say that [math]H\subset G[/math] is a normal subgroup.

\begin{proof} This is something well-known, the idea being as follows:

(1) The conditions in the statement are indeed equivalent, and in the classical case we obtain the usual normality notion for the subgroups.

(2) In the group dual case the normality of any subgroup, which must be a group dual subgroup, is then automatic, with this being something trivial.

(3) For more on these topics, and on the basic compact group theory in general, extended to the present quantum group setting, we refer to , . \end{proof} Summarizing, we have a quite complete theory for the notion of amenability, and for other related notions, coming either from discrete group theory, or from Lie theory.

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].