11d. Twisted geometry

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

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

There are many things that can be done in the context of the twisted geometry, going beyond what we have so far, namely some theory and computations for the spheres [math]\bar{S}[/math], tori [math]T[/math], unitary groups [math]\bar{U}[/math], and reflection groups [math]K[/math]. We briefly discuss here, as a main topic, the twisted extension of the various constructions from chapter 6.

So, let us go back to the theory there. As a first observation, we can both liberate the spaces [math]O_{MN}^L,U_{MN}^L[/math], and twist them, by proceeding as as follows:

Definition

Associated to any integers [math]L\leq M\leq N[/math] are the algebras

[[math]] \begin{eqnarray*} C(O_{MN}^{L+})&=&C^*\left((u_{ij})_{i=1,\ldots,M,j=1,\ldots,N}\Big|u=\bar{u},uu^t={\rm projection\ of\ trace}\ L\right)\\ C(U_{MN}^{L+})&=&C^*\left((u_{ij})_{i=1,\ldots,M,j=1,\ldots,N}\Big|uu^*,\bar{u}u^t={\rm projections\ of\ trace}\ L\right) \end{eqnarray*} [[/math]]

and their quotients [math]C(\bar{O}_{MN}^L),C(\bar{U}_{MN}^L)[/math], obtained by imposing the twisting relations.

With this extended formalism, we have inclusions between the various spaces constructed so far, in chapter 6 and then here, as follows:

[[math]] \xymatrix@R=15mm@C=15mm{ U_{MN}^L\ar[r]&U_{MN}^{L+}&\bar{U}_{MN}^L\ar[l]\\ O_{MN}^L\ar[r]\ar[u]&O_{MN}^{L+}\ar[u]&\bar{O}_{MN}^L\ar[l]\ar[u]} [[/math]]

More generally, we can perform these constructions for any quizzy quantum group. In order to discuss this, we use the Kronecker symbols [math]\delta_\pi(i)\in\{-1,0,1\}[/math], given by:

[[math]] \delta_\sigma(i)=\begin{cases} \delta_{\ker i\leq\sigma}&{\rm (untwisted\ case)}\\ \varepsilon(\ker i)\delta_{\ker i\leq\sigma}&{\rm (twisted\ case)} \end{cases} [[/math]]

With this convention, we have the following result, from :

Proposition

The various spaces [math]G_{MN}^L[/math] constructed so far appear by imposing to the standard coordinates of [math]U_{MN}^{L+}[/math] the relations

[[math]] \sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\delta_\pi(i)\delta_\sigma(j)u_{i_1j_1}^{e_1}\ldots u_{i_sj_s}^{e_s}=L^{|\pi\vee\sigma|} [[/math]]

with [math]s=(e_1,\ldots,e_s)[/math] ranging over all the colored integers, and with [math]\pi,\sigma\in D(0,s)[/math].

Show Proof

The relations defining [math]G_{MN}^L[/math] are as follows, with [math]\sigma[/math] ranging over a family of generators, with no upper legs, of the corresponding category of partitions [math]D[/math]:

[[math]] \sum_{j_1\ldots j_s}\delta_\sigma(j)u_{i_1j_1}^{e_1}\ldots u_{i_sj_s}^{e_s}=\delta_\sigma(i) [[/math]]

We therefore obtain the relations in the statement, as follows:

[[math]] \begin{eqnarray*} \sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\delta_\pi(i)\delta_\sigma(j)u_{i_1j_1}^{e_1}\ldots u_{i_sj_s}^{e_s} &=&\sum_{i_1\ldots i_s}\delta_\pi(i)\sum_{j_1\ldots j_s}\delta_\sigma(j)u_{i_1j_1}^{e_1}\ldots u_{i_sj_s}^{e_s}\\ &=&\sum_{i_1\ldots i_s}\delta_\pi(i)\delta_\sigma(i)\\ &=&\sum_{\tau\leq\pi\vee\sigma}\sum_{\ker i=\tau}(\pm1)^2\\ &=&\sum_{\tau\leq\pi\vee\sigma}\sum_{\ker i=\tau}1\\ &=&L^{|\pi\vee\sigma|} \end{eqnarray*} [[/math]]

As for the converse, this follows by using the relations in the statement, by keeping [math]\pi[/math] fixed, and by making [math]\sigma[/math] vary over all the partitions in the category.

■

Thus, we have unified the twisted and untwisted constructions, in the continuous case. In the general case now, where [math]G=(G_N)[/math] is an arbitary uniform quizzy quantum group, we can construct spaces [math]G_{MN}^L[/math] by using the above relations, and we have:

Theorem

The spaces [math]G_{MN}^L\subset U_{MN}^{L+}[/math] constructed by imposing the relations

[[math]] \sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\delta_\pi(i)\delta_\sigma(j)u_{i_1j_1}^{e_1}\ldots u_{i_sj_s}^{e_s}=L^{|\pi\vee\sigma|} [[/math]]

with [math]\pi,\sigma[/math] ranging over all the partitions in the associated category, having no upper legs, are subject to an action map/quotient map diagram, as follows,

[[math]] \xymatrix@R=15mm@C=30mm{ G\times G\ar[r]^m\ar[d]_{p\times id}&G\ar[d]^p\\ X\times G\ar[r]^a&X } [[/math]]

exactly as in the classical case, or the free case.

Show Proof

We proceed as in chapter 6. We must prove that, if the variables [math]u_{ij}[/math] satisfy the relations in the statement, then so do the following variables:

[[math]] U_{ij}=\sum_{kl}a_{ik}\otimes b_{jl}^*\otimes u_{kl} [[/math]]

[[math]] V_{ij}=\sum_{l\leq L}a_{il}\otimes b_{jl}^* [[/math]]

Regarding the variables [math]U_{ij}[/math], the computation here goes as follows:

[[math]] \begin{eqnarray*} &&\sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\delta_\pi(i)\delta_\sigma(j)U_{i_1j_1}^{e_1}\ldots U_{i_sj_s}^{e_s}\\ &=&\sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\sum_{k_1\ldots k_s}\sum_{l_1\ldots l_s}\delta_\pi(i)\delta_\sigma(j)a_{i_1k_1}^{e_1}\ldots a_{i_sk_s}^{e_s}\otimes(b_{j_sl_s}^{e_s}\ldots b_{j_1l_1}^{e_1})^*\otimes u_{k_1l_1}^{e_1}\ldots u_{k_sl_s}^{e_s}\\ &=&\sum_{k_1\ldots k_s}\sum_{l_1\ldots l_s}\delta_\pi(k)\delta_\sigma(l)u_{k_1l_1}^{e_1}\ldots u_{k_sl_s}^{e_s}\\ &=&L^{|\pi\vee\sigma|} \end{eqnarray*} [[/math]]

For the variables [math]V_{ij}[/math] the proof is similar, as follows:

[[math]] \begin{eqnarray*} &&\sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\delta_\pi(i)\delta_\sigma(j)V_{i_1j_1}^{e_1}\ldots V_{i_sj_s}^{e_s}\\ &=&\sum_{i_1\ldots i_s}\sum_{j_1\ldots j_s}\sum_{l_1,\ldots,l_s\leq L}\delta_\pi(i)\delta_\sigma(j)a_{i_1l_1}^{e_1}\ldots a_{i_sl_s}^{e_s}\otimes(b_{j_sl_s}^{e_s}\ldots b_{j_1l_1}^{e_1})^*\\ &=&\sum_{l_1,\ldots,l_s\leq L}\delta_\pi(l)\delta_\sigma(l)\\ &=&L^{|\pi\vee\sigma|} \end{eqnarray*} [[/math]]

Thus we have constructed an action map, and a quotient map, and the commutation of the diagram in the statement is then trivial.

■

Still by following the material in chapter 6, we can now construct a Haar integration for the above spaces, and we have the following result, also from :

Theorem

We have the Weingarten type formula

[[math]] \int_{G_{MN}^L}u_{i_1j_1}\ldots u_{i_sj_s}=\sum_{\pi\sigma\tau\nu}L^{|\sigma\vee\nu|}\delta_\pi(i)\delta_\tau(j)W_{sM}(\pi,\sigma)W_{sN}(\tau,\nu) [[/math]]

where [math]W_{sM}=G_{sM}^{-1}[/math], with [math]G_{sM}(\pi,\sigma)=M^{|\pi\vee\sigma|}[/math].

Show Proof

We make use of the usual quantum group Weingarten formula, explained in the above in the twisted case. By using this formula for [math]G_M,G_N[/math], we obtain:

[[math]] \begin{eqnarray*} \int_{G_{MN}^L}u_{i_1j_1}\ldots u_{i_sj_s} &=&\sum_{l_1\ldots l_s\leq L}\int_{G_M}a_{i_1l_1}\ldots a_{i_sl_s}\int_{G_N}b_{j_1l_1}^*\ldots b_{j_sl_s}^*\\ &=&\sum_{l_1\ldots l_s\leq L}\sum_{\pi\sigma}\delta_\pi(i)\delta_\sigma(l)W_{sM}(\pi,\sigma)\sum_{\tau\nu}\delta_\tau(j)\delta_\nu(l)W_{sN}(\tau,\nu)\\ &=&\sum_{\pi\sigma\tau\nu}\left(\sum_{l_1\ldots l_s\leq L}\delta_\sigma(l)\delta_\nu(l)\right)\delta_\pi(i)\delta_\tau(j)W_{sM}(\pi,\sigma)W_{sN}(\tau,\nu) \end{eqnarray*} [[/math]]

Let us compute now the coefficient appearing in the last formula. Since the signature map takes [math]\pm1[/math] values, for any multi-index [math]l=(l_1,\ldots,l_s)[/math] we have:

[[math]] \begin{eqnarray*} \delta_\sigma(l)\delta_\nu(l) &=&\delta_{\ker l\leq\sigma}\varepsilon(\ker l)\cdot\delta_{\ker l\leq\nu}\varepsilon(\ker l)\\ &=&\delta_{\ker l\leq\sigma\vee\nu} \end{eqnarray*} [[/math]]

Thus the coefficient is [math]L^{|\sigma\vee\nu|}[/math], and we obtain the formula in the statement.

■

With this formula in hand, we can derive explicit integration results for the sums of non-overlapping coordinates, exactly as in chapter 6. To be more precise, the laws and their asymptotics are identical in the classical and twisted cases. See .

General references

Banica, Teo (2024). "Affine noncommutative geometry". arXiv:2012.10973 [math.QA].