A simple explanation with types of the hexagonal moves of projective spaces

This post continues from the previous  A beautiful move in projective spaces    .

I call those “beautiful” moves hexagonal. So, how many hexagonal moves are?

In the post  Axioms for projective conical spaces (towards qubits II) I give 4 moves and I mention that I discard other two moves, because  don’t see their use in generalized projective geometry.

I was wrong, because in the usual projective geometry these two moves are discarded because they can be deduced from the barycentric move (which makes the generalized, “non-commutative”, projective geometry, into the usual one, in the same way as it makes the non-commutative affine geometry into the usual one). You really have to read the linked posts (and probably also the linked articles) in order to understand these precise statements. (So this is a kind of a filter for those with long attention span.)

The idea of using two types is natural: indeed, instead of using dual spaces, we use two types,  say “a” and “x”, and the decoration rule of the 4-valent dilation nodes:  two of the input arrows and the output arrow are decorated with the same type and the remaining input arrow is decorated with the other type. [ added: the “remaining input arrow” is always the one which points to the center of the circle which denoted the dilation node.]

By looking at the hexagonal moves from the last post, we see that there are only three ways of decorating the common part of all diagrams with types.  This gives 3 hexagonal moves.

This is explained in the next figure (click on it to make it bigger).

We see that some arrows are decorated with one type (arbitrarily called “x”) and other arrows are decorated, apparently with columns of 3 types. In fact, that should be read as 3 possibilities, which correspond to taking from each column the first, the second or the third element.

The choice corresponding to the first element of each column corresponds to the neglected hexagonal move, say (PG3). (Can you draw it? is easy!)

The choice corresponding to the second element of each column corresponds to (PG2).

The choice corresponding to the third element of each column corresponds to (PG1).

Remarking that the rule of decoration with types is symmetric with the switch between the types “a” and “x”, this corresponds to the 6 moves of generalized projective geometry.

In conclusion, by selecting only those graphs (and moves) which can be decorated in the mentioned way with two types, we get the moves of generalized projective geometry.

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A beautiful move in projective spaces

The post Axioms for projective conical spaces (towards qubits II)  introduces a generalization of projective spaces to projective conical space. These are a kind of non-commutative version of projective spaces, exactly in the same sense as the one that affine conical spaces are a non-commutative generalization of affine spaces.

That post has been done before the discovery of graphic lambda calculus. [UPDATE: no, I see that it was done after, but GLC was not used in that post.]

Now, the beautiful thing is that all the 4 axioms of projective conical spaces have the same form, if represented according to the same ideas as the ones of graphic lambda calculus.

There will be more about this, but I show you for the moment only how the first part of  (PG1)  looks like, in the original version and in the new version.

Here is the first part of (PG2) in old and new versions.

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Axioms for projective conical spaces (towards qubits II)

I am continuing from the post Towards qubits: graphic lambda calculus over conical groups and the barycentric move.  My goal here is to give a set of axioms for a “projective conical space”. Let me recall the following facts:

• affine conical spaces are the non-commutative equivalent of affine spaces. An affine conical space is constructed over a conical group as an affine space is constructed over a vector space.  Conical groups are generalizations of Carnot groups, in the sense that in the realm of Lie groups  the basic example of a conical group is a Carnot group. A conical Lie group is a contractive Lie group and therefore, by a theorem of Siebert, if it is simply connected then it is a nilpotent Lie group with a one-parameter family of contractive automorphisms. Carnot groups (think about examples as the Heisenberg group) are conical Lie groups with a supplementary hypothesis concerning the fact that the first level in the decomposition of the Lie algebra is generating the whole algebra.
• an affine  conical space is an usual affine space if and only if it satisfies the barycentric move. In this case and only in this case the underlying structure of the conical group is commutative.See  arXiv:0804.0135 [math.MG] for the introduction of “non-commutative affine geometry”, called here “affine conical geometry”, which generalizes results from W. Bertram  Generalized projective geometries: From linear algebra via affine algebra to projective algebra, Linear Algebra and its Applications 378 (2004), 109 – 134.
• afine conical spaces are defined in terms of a one-parameter family of quandle operations (called dilations). More specifically an affine conical space is generated by a one-parameter family of quandles which satisfy also some topological sugar axioms (which I’ll pass). More precisely, affine conical spaces are self-distributive uniform idempotent right quasigroups.  Uniform idempotent right quasigroups were introduced and studied under the shorter name “emergent algebras” in arXiv:0907.1520 [math.RA], see also   arXiv:1005.5031 [math.GR] for the context of studying them as algebraic-topologic generalizations of dilation structures (introduced in arXiv:math/0608536 [math.MG]), as well as for the description of symmetric spaces as emergent algebras.
• in  affine conical  geometry there is no notion of incidence or co-linearity, because of non-commutativity lurking beneath. However, there is a notion of a collinear triple of points, as well as a ratio associated to such points, but such collinear triples correspond to triples of   dilations (see further what “dilation” means) which, composed, give the identity. Such triples give the invariant of  affine conical geometry which corresponds to the ration of three collinear points in the usual affine geometry.

In the post Towards qubits I I explained (or linked to explanations) this in the language of graphic lambda calculus. Here I shall not use it fully, instead I shall use a graphical notation with variable names. But I think the correspondences between these two notations are rather clear. In particular I shall interpret identities as moves in trivalent graphs.

1. Algebraic axioms for affine conical spaces. (Topological sugar not included). We have a non-empty set $X$  and a commutative group of parameters $(\Gamma, \cdot, 1)$ with operation denoted multiplicatively $\cdot(\varepsilon, \mu) = \varepsilon \mu$ and neutral element $1$. Think about $\Gamma$ as being $(0,+\infty)$ or even $K^{*}$ where $K$ is a field.

On $X$  is defined a function $\delta: \Gamma \times X \times X \rightarrow X$ (Bertram uses the letter $\mu$ instead, I am using $\delta$). This function is to be interpreted as a $\Gamma$-parametrized family of operations. Namely we denote:

$\delta(\varepsilon, x, y) = \delta^{x}_{\varepsilon} y = x \circ_{\varepsilon} y$

This family of operations, called dilations, satisfies a number of algebraic axioms (as well as topological axioms which I pass), making them in particular into a family of quandle operations. I shall give these axioms in a graphical form, by using the transparent, I hope, notation:

Combinations (i.e. compositions) of dilations appear therefore as oriented trees with trivalent planar nodes decorated by the elements of $\Gamma$, with leaves (but not the root) decorated with elements from $X$.

The algebraic axioms of affine conical spaces are stating identities between certain compositions of dilations. Graphically these identities will be representes, as I wrote, as moves applied to such oriented trees.

Here are these axioms in graphical form:

(1) this  is equivalent with the move ext2    from graphical lambda calculus: (i.e. extensionality move 2)

(2) this is equivalent with the move R1a from graphical lambda calculus (i.e. Reidemeister move R1a, following the notation from Michael Polyak “Minimal generating sets of Reidemeister moves“)

(3) this is equivalent with the move R2 from the graphical lambda calculus (i.e. Reidemeister move 2, all Reidemeister moves 2 are equivalent in this formalism)

(4) this is the self-distributivity axiom, which could be called move R3b with the notations of Polyak

2. Algebraic axioms for projective conical spaces.  The intention is to propose a generalization of the same type, this time for projective spaces, of the one from W. Bertram Generalized projective geometries: General theory and equivalence with Jordan structures,  Advances in Geometry 3 (2002), 329-369.

This time we have a pair of spaces $(X,X')$. Think about the elements  $x \in X$ as being “points” and about the elements  $a \in X'$ as being “lines”, although, as in the case of affine conical geometry, there is no proper notion of incidence (except, of course, for the “commutative” particular case).

A pair geometry is a triple $(X,X',M)$ where $M \subset X \times X'$ is the set of pairs (say point-line) in general position. Compared to the more familiar case of incidence systems, the interpretation of $(x,a) \in M$ is “the point $x$ is not incident with the line $a$“.  The triple satisfies some conditions which I shall write after introducing some notations.

For any $x \in X$ and any $a \in A$ we denote:

$V_{x} = \left\{ b \in X' \mid (x,b) \in M \right\}$  and    $V_{a} = \left\{ y \in X \mid (y,a) \in M \right\}$.

Let also denote

$D = \left\{ (x,a,y) \in X \times X' \times X \mid (x,a), (y,a) \in M \right\}$ and $D' = \left\{ (a,x,b) \in X' \times X \times X' \mid (x,a), (x,b) \in M \right\}$.

(Pair geometry 1) for any $x \in X$ and for any $a \in X'$ the sets $V_{x}$ and $V_{a}$ are non-empty,

(Pair geometry 2) for any pair of different points $x,y \in X$ there exists and it’s unique a line $a \in X'$ such that $(x,a)$ and $(y,a)$ are not in $M$; dually, for any pair of different lines $a,b \in X'$ there exists and it’s unique a point $x \in X$ such that $(x,a)$ and $(x,b)$ are not in $M$.

Remark. This is the definition of a pair geometry given by Bertram. I shall keep further only (Pair geometry 1) because I feel that (Pair geometry 2) has too much “incidence content” which might be not non-commutative enough. So, for the moment, (Pair geometry 2) is in quarantine. As a first suggestion coming into mind, it might well turn out that it can be replaced by a more lax version saying that there is a number $N$ such that $X$ is covered by the reunion of $N$  sets $V_{x}$ (and a similar dual formulation for $X'$. As it is, (Pair geometry 2) corresponds to such a formulation for $N = 3$.

We want the following:

1. for any point $x \in X$ the space $V_{x}$ is an affine conical space,
2. for any line $a \in X'$ the space $V_{a}$ is an affine conical space,
3. these structures are glued together by some axioms.

Let’s pass through these three points of the list.

1.  that means we shall put a structure of dilation operations on every $V_{x}$. It is natural then to mark the dilation operations not only by elements of the group $\Gamma$, but also by $x$. More concretely that means we introduce for any $\varepsilon \in \Gamma$  a function

$\delta_{\varepsilon}: D' \rightarrow X'$

which, for any $x \in X$ it takes a pair of lines $(a,b)$, with $a,b \in V_{x}$ and returns $\delta_{\varepsilon}(a,x,b) \in V_{x}$.

We ask that for any $x \in X$ the dilations $\varepsilon \mapsto \delta_{\varepsilon}(\cdot, x, \cdot)$ satisfy axioms (1), (2), (3) of affine conical spaces.

2. in the same way, we want that every $V_{a}$ to have a structure of dilation operations. We have therefore, for any $\varepsilon \in \Gamma$ another function (but I shall use the same letter $\delta$)

$\delta_{\varepsilon}: D \rightarrow X$

which, for any $a \in X'$ it takes a pair of points $(x,y)$, with $x,y \in V_{a}$ and returns $\delta_{\varepsilon}(x,a,y) \in V_{a}$.

We ask that for any $a \in X'$ the dilations $\varepsilon \mapsto \delta_{\varepsilon}(\cdot, a, \cdot)$ satisfy axioms (1), (2), (3) of affine conical spaces.

3.  the gluing axioms are generalizations of axioms (PG1), (PG2) of Bertram. In the mentioned article, Bertram explains that these two axioms lead to eight identities. From those eight, six of them are different. From those six, Bertram is using the barycentric axiom to eliminate two of them, which leaves him with four identities. I shall not use the barycentric axiom, because otherwise I shall fall on the commutative case, but  I shall eliminate as well  these two axioms> Therefore I shall have  four  moves which will replace the Reidemeister move 3 axiom , i.e. the self-distributivity move (4) from affine conical spaces.

Remark. Bertram adds some sugar over (PG1) and (PG2) which serves to be able to construct tangent structures further. I renounce at those in favor of  my topological sugar which I pass, for the moment.

Remark. As we saw that the axioms of affine conical spaces are practically corresponding to the Reidemeister moves, it is natural to expect that the four  axioms correspond to either: the Roseman moves, or to some 2-quandle definition. I need help and suggestions here!

I shall write further the four axioms which replace the axiom (4), that is why I shall name them (4.1) … (4.4). As previously I shall use a graphical notation, which my visual brain finds more easy to understand than the notation using multiple compositions of functions with 4 arguments (however, see Bertram’s notations involving adjoint pairs). Also, there are limits to my capacity to write latex formulae which are well parsed in this blog.

So, here is the notation for dilations which I shall use for writing those four axioms:

Let’s look at the first line. For any $a \in X'$ we have an associated dilation operation taking as input a pair of points $x,y \in V_{a}$. Graphically this is represented by a node with two inputs and an output, together with a planar embedding  (i.e. the local planar embedding tells us which is the left input and which is the right output), and  with a supplementary input which points to the center of the circle (node), serving to identify the node as the dilation in the space $V_{a}$. Similar comments could be made about the second line of the figure.

Therefore, this time we are working with trees made by 4-valent nodes, each node having three inputs and one output and moreover with a triple of two inputs and the output with an orientation given.  The leaves, but not the root of such a tree are decorated by points or lines. There should be other constraints on this family of trees, coming from the fact that if the input which points to the center of the circle correspond to a point then the other inputs should correspond to lines, and so on. For the moment I pass over this, probably a solution would be to colour the edges, by using two colors, one for points, the other for lines, then express the constraints in terms of those colors.

As previously, the nodes are decorated by elements of the commutative group $\Gamma$.

(4.1)     first part of (PG1)

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(4.2) second part of (PG1)

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(4.3)  first  part of (PG2)

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(4.4) second part of (PG.2)

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In a future post I shall give:

• a theorem of characterization of projective conical spaces, of the same type as the theorem of characterization of affine conical spaces
• examples of non-commutative projective conical spaces, in particular answering to the question: what is the natural notion of a projective space of a conical group (more particularly, if we think about Carnot groups as being non-commutative vector spaces, then who are their associated non-commutative projective spaces?).

UPDATE:  The axioms (4.1) … (4.4) take a much more simple form if we use choroi and differences, but that’s also for a future post.

Towards qubits: graphic lambda calculus over conical groups and the barycentric move

In this post I want to pave the way to the application of graphic lambda calculus to the realm of quantum computation. It is not a short, nor too lengthy way, which will be explained in several posts. Also, some experimentation is to be expected.

Disclaimer: For the moment it is not very clear to me which are the exact relations between the approach I am going to explain and linear lambda calculus or the  lambda calculus for quantum computation.  I expect a certain overlapping, but maybe not as much as expected (by the specialist in the field). The reason is that the instruments and goals which I have come from fields apparently far away from quantum computation, as for example sub-riemannian geometry, which is my main field of interest (however, for an interaction between sub-riemannian geometry and computation  see  L_p metrics on the Heisenberg group and the Goemans-Linial conjecture, by James R. Lee and Asaf Naor). Therefore, I feel the need to issue such a disclaimer for the narrow specialist.

Background for his post:

• The page Graphic lambda calculus
•  [1] Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston J.  Math., 36, 1 (2010), 91-136, arXiv:0804.0135.
• [2]  On graphic lambda calculus and the dual of the graphic beta move, arXiv:1302.0778.

Affine conical spaces. In the article [1] they appear under the name “normed affine group spaces”, definition 3. We may use the same type of arguments as the ones from emergent algebras   in order to get rid of the need to have a norm on such spaces.  Instead of anorm we shall put an uniformity on such a space, such that the topology associated to the uniformity makes the space to be locally compact.

Theorem 2.2 [1] characterizes affine conical spaces as self-distributive emergent algebras. The relations satisfied by self-distributive emergent algebras, if graphically represented by gates in graphic lambda calculus, are the following:

Notice that I don’t want to use the dual of the graphic beta move ([2], section 8), which is simply too powerful in this context (see [2] section 10). That is why I use instead the move R3a (which is a composite of dual beta moves). Another instance of this choice will be explained in a future post, having to do with the distributivity of the emergent algebra operations with respect to the application and lambda gates.

The barycentric move.   In order to obtain usual affine spaces instead of their more general,  noncommutative versions (i.e. affine conical spaces), we have to add the barycentric condition. This condition appears as (Af3) in Theorem 2.2 [1].  I shall transform this condition into a move in graphic lambda calculus.

The barycentric move BAR is described by the following figure and explanation. We take the commutative group $\Gamma$, which is used to label the emergent algebra gates, as $\Gamma = K^{*}$, where $K$ is a field. (Therefore $K = \Gamma \cup \left\{ 0 \right\}$.) We have then two operations on the field $K$: multiplication $\varepsilon, \mu) \mapsto \varepsilon \mu$ and addition $(\varepsilon, \mu) \mapsto \varepsilon + \mu$. Because $K$ contains also the element $0$, the neutral element for addition, we add a new gate $\bar{0}$.  With these preparations, the BAR move is the following:

Notice that when $\varepsilon = 1$ at the left hand side of the figure appears the gate $\bar{0}$. This gate corresponds, in the particular case of a vector space, to the usual dilation of coefficient $0$. We don’t need to put this as a sort of an axiom, because we can obtain it as a combination of the BAR move and ext2 moves. Indeed:

Knowing this, we can extend the emergent algebra moves R1a, R1b and R2 to the case $\varepsilon = 0$. Here is the proof. For R1a we do this:

The move R1b, for the degenerate case $\varepsilon = 0$, is this:

Finally, for the move R2 we have two cases, corresponding to $0 \, \varepsilon = 0$ and $\varepsilon \, 0 = 0$. The first case is this:

The second case is this:

Final remark: The move BAR can be seen as analogous of an infinite sequence of moves R3 (but there is no rigorous sense for this in graphic lambda calculus). Indeed, this is related to the fact that $\frac{1}{1-\varepsilon} = \sum^{\infty}_{0} \varepsilon^{k}$.  See [1] section 8 “Noncommutative affine geometry” for the dilation structures correspondent of this equality and also see the post Menelaus theorem by way of Reidemeister move 3.