2c. Product operations

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

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We have seen so far that the compact quantum Lie groups can be axiomatized, and that as a bonus, we obtain in this way a definition as well for the finitely generated discrete quantum groups. Let us get now into a more exciting question, namely the construction of examples. We first have the following construction, due to Wang :

Proposition

Given two compact quantum groups [math]G,H[/math], so is their product [math]G\times H[/math], constructed according to the following formula:

[[math]] C(G\times H)=C(G)\otimes C(H) [[/math]]

Equivalently, at the level of the associated discrete duals [math]\Gamma,\Lambda[/math], we can set

[[math]] C^*(\Gamma\times\Lambda)=C^*(\Gamma)\otimes C^*(\Lambda) [[/math]]

and we obtain the same equality of Woronowicz algebras as above.

Show Proof

Assume indeed that we have two Woronowicz algebras, [math](A,u)[/math] and [math](B,v)[/math]. Our claim is that the following construction produces a Woronowicz algebra:

[[math]] C=A\otimes B\quad,\quad w=diag(u,v) [[/math]]

Indeed, the matrix [math]w[/math] is unitary, and its coefficients generate [math]C[/math]. As for the existence of the maps [math]\Delta,\varepsilon,S[/math], this follows from the functoriality properties of [math]\otimes[/math], which is here, as usual, the universal [math]C^*[/math]-algebraic completion of the algebraic tensor product. With this claim in hand, the first assertion is clear. As for the second assertion, let us recall that when [math]G,H[/math] are classical and abelian, we have the following formula:

[[math]] \widehat{G\times H}=\widehat{G}\times\widehat{H} [[/math]]

Thus, our second assertion is simply a reformulation of the first assertion, with the [math]\times[/math] symbol used there being justified by this well-known group theory formula.

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Here is now a more subtle construction, once again due to Wang :

Proposition

Given two compact quantum groups [math]G,H[/math], so is their dual free product [math]G\,\hat{*}\,H[/math], constructed according to the following formula:

[[math]] C(G\,\hat{*}\,H)=C(G)*C(H) [[/math]]

Equivalently, at the level of the associated discrete duals [math]\Gamma,\Lambda[/math], we can set

[[math]] C^*(\Gamma*\Lambda)=C^*(\Gamma)*C^*(\Lambda) [[/math]]

and we obtain the same equality of Woronowicz algebras as above.

Show Proof

The proof here is identical with the proof of Proposition 2.17, by replacing everywhere the tensor product [math]\otimes[/math] with the free product [math]*[/math], with this latter product being by definition the universal [math]C^*[/math]-algebraic completion of the algebraic free product.

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Here is another construction, which once again, has no classical counterpart:

Proposition

Given a compact quantum group [math]G[/math], so is its free complexification [math]\widetilde{G}[/math], constructed according to the following formula, where [math]z=id\in C(\mathbb T)[/math]:

[[math]] C(\widetilde{G})\subset C(\mathbb T)*C(G)\quad,\quad\tilde{u}=zu [[/math]]

Equivalently, at the level of the associated discrete dual [math]\Gamma[/math], we can set

[[math]] C^*(\widetilde{\Gamma})\subset C^*(\mathbb Z)*C^*(\Gamma)\quad,\quad\tilde{u}=zu [[/math]]

where [math]z=1\in\mathbb Z[/math], and we obtain the same Woronowicz algebra as above.

Show Proof

This follows from Proposition 2.17. Indeed, we know that [math]C(\mathbb T)*C(G)[/math] is a Woronowicz algebra, with matrix of coordinates [math]w=diag(z,u)[/math]. Now, let us try to replace this matrix with the matrix [math]\tilde{u}=zu[/math]. This matrix is unitary, and we have:

[[math]] \Delta(\tilde{u}_{ij}) =(z\otimes z)\sum_ku_{ik}\otimes u_{kj} =\sum_k\tilde{u}_{ik}\otimes\tilde{u}_{kj} [[/math]]

Similarly, in what regards the counit, we have the following formula:

[[math]] \varepsilon(\tilde{u}_{ij}) =1\cdot\delta_{ij} =\delta_{ij} [[/math]]

Finally, recalling that [math]S[/math] takes values in the opposite algebra, we have as well:

[[math]] S(\tilde{u}_{ij}) =u_{ji}^*\cdot\bar{z} =\tilde{u}_{ji}^* [[/math]]

Summarizing, the conditions in Definition 2.8 are satisfied, except for the fact that the entries of [math]\tilde{u}=zu[/math] do not generate the whole algebra [math]C(\mathbb T)*C(G)[/math]. We conclude that if we let [math]C(\widetilde{G})\subset C(\mathbb T)*C(G)[/math] be the subalgebra generated by the entries of [math]\tilde{u}=zu[/math], as in the statement, then the conditions in Definition 2.8 are satisfied, as desired.

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Another standard operation is that of taking subgroups:

Proposition

Let [math]G[/math] be compact quantum group, and let [math]I\subset C(G)[/math] be a closed [math]*[/math]-ideal satisfying the following condition:

[[math]] \Delta(I)\subset C(G)\otimes I+I\otimes C(G) [[/math]]

We have then a closed quantum subgroup [math]H\subset G[/math], constructed as follows:

[[math]] C(H)=C(G)/I [[/math]]

At the dual level we obtain a quotient of discrete quantum groups, [math]\widehat{\Gamma}\to\widehat{\Lambda}[/math].

Show Proof

This follows indeed from the above conditions on [math]I[/math], which are designed precisely as for [math]\Delta,\varepsilon,S[/math] to factorize through the quotient. As for the last assertion, this is just a reformulation, coming from the functoriality properties of the Pontrjagin duality.

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In order to discuss now the quotient operation, let us agree to call “corepresentation” of a Woronowicz algebra [math]A[/math] any unitary matrix [math]v\in M_n(\mathcal A)[/math] satisfying:

[[math]] \Delta(v_{ij})=\sum_kv_{ik}\otimes v_{kj}\quad,\quad \varepsilon(v_{ij})=\delta_{ij}\quad,\quad S(v_{ij})=v_{ji}^* [[/math]]

We will study in detail such corepresentations in chapter 3 below. For the moment, we just need their definition, in order to formulate the following result:

Proposition

Let [math]G[/math] be a compact quantum group, and [math]v=(v_{ij})[/math] be a corepresentation of [math]C(G)[/math]. We have then a quotient quantum group [math]G\to H[/math], given by:

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

At the dual level we obtain a discrete quantum subgroup, [math]\widehat{\Lambda}\subset\widehat{\Gamma}[/math].

Show Proof

Here the first assertion follows from the above definition of the corepresentations, and the second assertion is just a reformulation of it, coming from the basic functoriality properties of the Pontrjagin duality.

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Finally, here is one more construction, which is something more tricky, and which will be of importance in what follows:

Theorem

Given a compact quantum group [math]G[/math], with fundamental corepresentation denoted [math]u=(u_{ij})[/math], the [math]N^2\times N^2[/math] matrix given in double index notation by

[[math]] v_{ia,jb}=u_{ij}u_{ab}^* [[/math]]

is a corepresentation in the above sense, and we have the following results:

• The corresponding quotient [math]G\to PG[/math] is a compact quantum group.
• Via the standard embedding [math]G\subset S^{N^2-1}_{\mathbb C,+}[/math], this is the projective version.
• In the classical group case, [math]G\subset U_N[/math], we have [math]PG=G/(G\cap\mathbb T^N)[/math].
• In the group dual case, with [math]\Gamma= \lt g_i \gt [/math], we have [math]\widehat{P\Gamma}= \lt g_ig_j^{-1} \gt [/math].

Show Proof

The fact that [math]v[/math] is indeed a corepresentation is routine, and follows as well from the general properties of such corepresentations, to be discussed in chapter 3 below. Regarding now other assertions, the proofs go as follows:

(1) This follows from Proposition 2.21 above.

(2) Observe first that, since the matrix [math]v=(v_{ia,jb})[/math] is biunitary, we have indeed an embedding [math]G\subset S^{N^2-1}_{\mathbb C,+}[/math] as in the statement, given in double index notation by [math]x_{ia,jb}=\frac{v_{ia,jb}}{N}[/math]. Now with this formula in hand, the assertion is clear from definitions.

(3) This follows from the elementary fact that, via Gelfand duality, [math]w[/math] is the matrix of coefficients of the adjoint representation of [math]G[/math], whose kernel is the subgroup [math]G\cap\mathbb T^N[/math], where [math]\mathbb T^N\subset U_N[/math] denotes the subgroup formed by the diagonal matrices.

(4) This is something trivial, which follows from definitions.

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General references

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