12d. Half-liberation

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

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As a nice illustration for the above modelling theory, let us discuss now the half-liberation operation, which is connected to [math]X^{(2)}[/math], as a continuation of the material from chapter 9. We first restrict the attention to the real case. Let us start with:

Definition

The half-classical version of a manifold [math]X\subset S^{N-1}_{\mathbb R,+}[/math] is given by:

[[math]] C(X^*)=C(X)\Big/\left \lt abc=cba\Big|\forall a,b,c\in\{x_i\}\right \gt [[/math]]

We say that [math]X[/math] is half-classical when [math]X=X^*[/math].

Observe the obvious similarity with the construction of the classical version. In fact, philosophically, this definition is some sort of “next level” definition for the classical version, assuming that you managed, via some sort of yoga, to be as familiar with half-commutation, [math]abc=cba[/math], as you are with usual commutation, [math]ab=ba[/math].

In order to understand now the structure of [math]X^*[/math], we can use an old matrix model method, which goes back to Bichon-Dubois-Violette , and then to Bichon . This is based on the following observation, that we already met in chapter 9:

Proposition

For any [math]z\in\mathbb C^N[/math], the matrices

[[math]] X_i=\begin{pmatrix}0&z_i\\ \bar{z}_i&0\end{pmatrix} [[/math]]

are self-adjoint, and half-commute.

Show Proof

The matrices [math]X_i[/math] are indeed self-adjoint, and their products are given by:

[[math]] X_iX_j =\begin{pmatrix}0&z_i\\ \bar{z}_i&0\end{pmatrix}\begin{pmatrix}0&z_j\\ \bar{z}_j&0\end{pmatrix} =\begin{pmatrix}z_i\bar{z}_j&0\\ 0&\bar{z}_iz_j\end{pmatrix} [[/math]]

Also, we have as well the following formula:

[[math]] X_iX_jX_k =\begin{pmatrix}z_i\bar{z}_j&0\\ 0&\bar{z}_iz_j\end{pmatrix}\begin{pmatrix}0&z_k\\ \bar{z}_k&0\end{pmatrix} =\begin{pmatrix}0&z_i\bar{z}_jz_k\\ \bar{z}_iz_j\bar{z}_k&0\end{pmatrix} [[/math]]

Now since this latter quantity is symmetric in [math]i,k[/math], we obtain from this that we have the half-commutation formula [math]X_iX_jX_k=X_kX_jX_i[/math], as desired.

■

The idea now, following Bichon-Dubois-Violette and Bichon , will be that of using the matrices in Proposition 12.29 in order to model the coordinates of the arbitrary half-classical manifolds. In order to connect the algebra of the classical coordinates [math]z_i[/math] to that of the noncommutative coordinates [math]X_i[/math], we will need an abstract definition:

Definition

Given a noncommutative polynomial [math]f\in\mathbb R \lt x_1,\ldots,x_N \gt [/math] in [math]N[/math] variables, we define a usual polynomial in [math]2N[/math] variables

[[math]] f^\circ\in\mathbb R[z_1,\ldots,z_N,\bar{z}_1,\ldots,\bar{z}_N] [[/math]]

according to the formula

[[math]] f=x_{i_1}x_{i_2}x_{i_3}x_{i_4}\ldots\implies f^\circ=z_{i_1}\bar{z}_{i_2}z_{i_3}\bar{z}_{i_4}\ldots [[/math]]

in the monomial case, and then by extending this correspondence, by linearity.

As a basic example here, the polynomial defining the free real sphere [math]S^{N-1}_{\mathbb R,+}[/math] produces in this way the polynomial defining the complex sphere [math]S^{N-1}_\mathbb C[/math]:

[[math]] f=x_1^2+\ldots+x_N^2\implies f^\circ=|z_1|^2+\ldots+|z_N|^2 [[/math]]

Also, given a polynomial [math]f\in\mathbb R \lt x_1,\ldots,x_N \gt [/math], we can decompose it into its even and odd parts, [math]f=g+h[/math], by putting into [math]g/h[/math] the monomials of even/odd length. Observe that with [math]z=(z_1,\ldots,z_N)[/math], these odd and even parts are given by:

[[math]] g(z)=\frac{f(z)+f(-z)}{2}\quad,\quad h(z)=\frac{f(z)-f(-z)}{2} [[/math]]

With these conventions, we have the following result:

Proposition

Given a manifold [math]X[/math], coming from a family of noncommutative polynomials [math]\{f_\alpha\}\subset\mathbb R \lt x_1,\ldots,x_N \gt [/math], we have a morphism algebras

[[math]] \pi:C(X)\to M_2(\mathbb C)\quad,\quad \pi(x_i)=\begin{pmatrix}0&z_i\\ \bar{z}_i&0\end{pmatrix} [[/math]]

precisely when [math]z=(z_1,\ldots,z_N)\in\mathbb C^N[/math] belongs to the real algebraic manifold

[[math]] Y=\left\{z\in\mathbb C^N\Big|g^\circ_\alpha(z_1,\ldots,z_N)=h^\circ_\alpha(z_1,\ldots,z_N)=0,\forall\alpha\right\} [[/math]]

where [math]f_\alpha=g_\alpha+h_\alpha[/math] is the even/odd decomposition of [math]f_\alpha[/math].

Show Proof

Let [math]X_i[/math] be the matrices in the statement. In order for [math]x_i\to X_i[/math] to define a morphism of algebras, these matrices must satisfy the equations defining [math]X[/math]. Thus, the space [math]Y[/math] in the statement consists of the points [math]z=(z_1,\ldots,z_N)\in\mathbb C^N[/math] satisfying:

[[math]] f_\alpha(X_1,\ldots,X_N)=0\quad,\quad\forall\alpha [[/math]]

Now observe that the matrices [math]X_i[/math] in the statement multiply as follows:

[[math]] X_{i_1}X_{j_1}\ldots X_{i_k}X_{j_k}=\begin{pmatrix}z_{i_1}\bar{z}_{j_1}\ldots z_{i_k}\bar{z}_{j_k}&0\\ 0&\bar{z}_{i_1}z_{j_1}\ldots\bar{z}_{i_k}z_{j_k}\end{pmatrix} [[/math]]

[[math]] X_{i_1}X_{j_1}\ldots X_{i_k}X_{j_k}X_{i_{k+1}}=\begin{pmatrix}0&z_{i_1}\bar{z}_{j_1}\ldots z_{i_k}\bar{z}_{j_k}z_{i_{k+1}}\\ \bar{z}_{i_1}z_{j_1}\ldots\bar{z}_{i_k}z_{j_k}\bar{z}_{i_{k+1}}&0\end{pmatrix} [[/math]]

We therefore obtain, in terms of the even/odd decomposition [math]f_\alpha=g_\alpha+h_\alpha[/math]:

[[math]] f_\alpha(X_1,\ldots,X_N)=\begin{pmatrix}g^\circ_\alpha(z_1,\ldots,z_N)&h^\circ_\alpha(z_1,\ldots,z_N)\\ \\ \overline{h^\circ_\alpha(z_1,\ldots,z_N)}&\overline{g^\circ_\alpha(z_1,\ldots,z_N)}\end{pmatrix} [[/math]]

Thus, we obtain the equations for [math]Y[/math] from the statement.

■

As a first consequence, of theoretical interest, a necessary condition for [math]X[/math] to exist is that the manifold [math]Y\subset\mathbb C^N[/math] constructed above must be compact, and we will be back to this later. In order to discuss now modelling questions, we will need as well:

Definition

Assuming that we are given a manifold [math]Z[/math], appearing via

[[math]] C(Z)=C^*\left(z_1,\ldots,z_N\Big|f_\alpha(z_1,\ldots,z_N)=0\right) [[/math]]

we define the projective version of [math]Z[/math] to be the quotient space [math]Z\to PZ[/math] corresponding to the subalgebra [math]C(PZ)\subset C(Z)[/math] generated by the variables [math]x_{ij}=z_iz_j^*[/math].

The relation with the half-classical manifolds comes from the fact that the projective version of a half-classical manifold is classical. Indeed, from [math]abc=cba[/math] we obtain:

[[math]] \begin{eqnarray*} ab\cdot cd &=&(abc)d\\ &=&(cba)d\\ &=&c(bad)\\ &=&c(dab)\\ &=&cd\cdot ab \end{eqnarray*} [[/math]]

Finally, let us call as before “matrix model” any morphism of unital [math]C^*[/math]-algebras [math]f:A\to B[/math], with target algebra [math]B=M_K(C(Y))[/math], with [math]K\in\mathbb N[/math], and [math]Y[/math] being a compact space. With these conventions, following Bichon , we have the following result:

Theorem

Given a half-classical manifold [math]X[/math] which is symmetric, in the sense that all its defining polynomials [math]f_\alpha[/math] are even, its universal [math]2\times2[/math] antidiagonal model,

[[math]] \pi:C(X)\to M_2(C(Y)) [[/math]]

where [math]Y[/math] is the manifold constructed in Proposition 12.31, is faithful. In addition, the construction [math]X\to Y[/math] is such that [math]X[/math] exists precisely when [math]Y[/math] is compact.

Show Proof

We can proceed as in . Indeed, the universal model [math]\pi[/math] in the statement induces, at the level of projective versions, a certain representation:

[[math]] C(PX)\to M_2(C(PY)) [[/math]]

By using the multiplication formulae from the proof of Proposition 12.31, the image of this representation consists of diagonal matrices, and the upper left components of these matrices are the standard coordinates of [math]PY[/math]. Thus, we have an isomorphism:

[[math]] PX\simeq PY [[/math]]

We can conclude then by using a grading trick. See .

■

As a first observation, this result shows that when [math]X[/math] is symmetric, we have [math]X^*\subset X^{(2)}[/math]. Going beyond this observation is an interesting problem.

In what follows, we will rather need a more detailed version of the above result. For this purpose, we can use the following definition:

Definition

Associated to any compact manifold [math]Y\subset\mathbb C^N[/math] is the real compact half-classical manifold [math][Y][/math], having as coordinates the following variables,

[[math]] X_i=\begin{pmatrix}0&z_i\\ \bar{z}_i&0\end{pmatrix} [[/math]]

where [math]z_1,\ldots,z_N[/math] are the standard coordinates on [math]Y[/math]. In other words, [math][Y][/math] is given by the fact that [math]C([Y])\subset M_2(C(Y))[/math] is the algebra generated by these matrices.

Here the fact that the manifold [math][Y][/math] is indeed half-classical follows from the results above. As for the fact that [math][Y][/math] is indeed algebraic, this follows from Theorem 12.33. Now with this notion in hand, we can reformulate Theorem 12.33, as follows:

Theorem

The symmetric half-classical manifolds [math]X[/math] appear as follows:

• We have [math]X=[Y][/math], for a certain conjugation-invariant subspace [math]Y\subset \mathbb C^N[/math].
• [math]PX=P[Y][/math], and [math]X[/math] is maximal with this property.
• In addition, we have an embedding [math]C([X])\subset C(X)\rtimes\mathbb Z_2[/math].

Show Proof

This follows from Theorem 12.33, with the embedding in (3) being constructed as in , by [math]x_i=z_i\otimes\tau[/math], where [math]\tau[/math] is the standard generator of [math]\mathbb Z_2[/math]. See .

■

And this is all, on this subject. In the unitary case things are a bit more complicated, and in connection with this, there are also some higher analogues of the above developed, using [math]K\times K[/math] matrix models. We refer to , , for more on these topics.

As a conclusion now, and by getting back to the real case, for simplifying, the half-classical geometry can be normally developed in a quite efficient way, at a technical level which is close to that of the classical one, by using [math]2\times2[/math] matrix models, as indicated above. Of course, all this still remains to be done. In fact, as already mentioned in chapter 9 and afterwards, on several occasions, there are plenty of interesting things to be done here, and there is certainly room for writing a nice book on the subject.

Which reminds a bit the situation with the twisting, from chapter 11, with a nice book to be written there as well. In fact, both the half-classical geometry and the twisted geometry, and their combination the half-classical twisted geometry, which is something which exists as well, are somehow examples of “tame geometries”, not far from the classical geometry, and with a bewildering array of techniques, including those of Connes , potentially applying, and with very interesting results at stake.

And a word about physics, to finish with. Although there is nothing much concrete here, at least so far, a quite common belief is that, mathematically speaking somehow, QED is supposed to be something tame, and QCD is supposed to be something wild. And this is why we have mixed tame and wild things in this book, with tame and wild meaning for us something purely mathematical, namely non-free and free, with the belief that things are in correspondence, and that all this can be of help in physics.

We will be back to more speculations in chapters 13-16 below, when discussing more in detail free geometry, in continuation of the material from chapters 5-8, and benefiting too from what we learned from here, chapters 9-12, now coming to an end.

General references

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