# 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)

______________________

(4.2) second part of (PG1)

______________________

(4.3)  first  part of (PG2)

______________________

(4.4) second part of (PG.2)

______________________

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.

# Noncommutative Baker-Campbell-Hausdorff formula: the problem

I come back to a problem alluded in a previous post, where the proof of the Baker-Campbell-Hausdorff formula from this post by Tao is characterized as “commutative”, because of the “radial homogeneity” condition in his Theorem 1 , which forces commutativity.

Now I am going to try to explain this, as well as what the problem of a “noncommutative” BCH formula would be.

Take a Lie group $G$ and identify a neighbourhood of its neutral element with a neighbourhood of the $0$ element of its Lie algebra. This is standard for Carnot groups (connected, simply connected nilpotent groups which admit a one parameter family of contracting automorphisms), where the exponential is bijective, so the identification is global. The advantage of this identification is that we get rid of log’s and exp’s in formulae.

For every $s > 0$ define a deformation of the group operation (which is denoted multiplicatively), by the formula

(1)                $s(x *_{s} y) = (sx) (sy)$

Then we have $x *_{s} y \rightarrow x+y$ as $s \rightarrow 0$.

Denote by $[x,y]$ the Lie bracket of the (Lie algebra of the) group $G$ with initial operation and likewise denote by $[x,y]_{s}$ the Lie bracket of the operation $*_{s}$.

The relation between these brackets is: $[x,y]_{s} = s [x,y]$.

From the Baker-Campbell-Hausdorff formula we get:

$-x + (x *_{s} y) - y = \frac{s}{2} [x,y] + o(s)$,

(for reasons which will be clear later, I am not using the commutativity of addition), therefore

(2)         $\frac{1}{s} ( -x + (x *_{s} y) - y ) \rightarrow \frac{1}{2} [x,y]$       as        $s \rightarrow 0$.

Remark that (2) looks like a valid definition of the Lie bracket which is not related to the group commutator. Moreover, it is a formula where we differentiate only once, so to say. In the usual derivation of the Lie bracket from the group commutator we have to differentiate twice!

Let us now pass to a slightly different context: suppose $G$ is a normed group with dilations (the norm is for simplicity, we can do without; in the case of “usual” Lie groups, taking a norm corresponds to taking a left invariant Riemannian distance on the group).

$G$ is a normed group with dilations if

• it is a normed group, that is there is a norm function defined on $G$ with values in $[0,+\infty)$, denoted by $\|x\|$, such that

$\|x\| = 0$ iff $x = e$ (the neutral element)

$\| x y \| \leq \|x\| + \|y\|$

$\| x^{-1} \| = \| x \|$

– “balls” $\left\{ x \mid \|x\| \leq r \right\}$ are compact in the topology induced by the distance $d(x,y) = \|x^{-1} y\|$,

• and a “multiplication by positive scalars” $(s,x) \in (0,\infty) \times G \mapsto sx \in G$ with the properties:

$s(px) = (sp)x$ , $1x = x$ and $sx \rightarrow e$ as $s \rightarrow 0$; also $s(x^{-1}) = (sx)^{-1}$,

– define $x *_{s} y$ as previously, by the formula (1) (only this time use the multiplication by positive scalars). Then

$x *_{s} y \rightarrow x \cdot y$      as      $s \rightarrow 0$

uniformly with respect to $x, y$ in an arbitrarry closed ball.

$\frac{1}{s} \| sx \| \rightarrow \|x \|_{0}$, uniformly with respect to $x$ in a closed ball, and moreover $\|x\|_{0} = 0$ implies $x = e$.

1. In truth, everything is defined in a neighbourhood of the neutral element, also $G$ has only to be a local group.

2. the operation $x \cdot y$ is a (local) group operation and the function $\|x\|_{0}$ is a norm for this operation, which is also “homogeneous”, in the sense

$\|sx\|_{0} = s \|x\|_{0}$.

Also we have the distributivity property $s(x \cdot y) = (sx) \cdot (sy)$, but generally the dot operation is not commutative.

3. A Lie group with a left invariant Riemannian distance $d$ and with the usual multiplication by scalars (after making the identification of a neighbourhood of the neutral element with a neighbourhood in the Lie algebra) is an example of a normed group with dilations, with the norm $\|x\| = d(e,x)$.

4. Any Carnot group can be endowed with a structure of a group with dilations, by defining the multiplication by positive scalars with the help of its intrinsic dilations. Indeed, take for example a Heisenberg group $G = \mathbb{R}^{3}$ with the operation

$(x_{1}, x_{2}, x_{3}) (y_{1}, y_{2}, y_{3}) = (x_{1} + y_{1}, x_{2} + y_{2}, x_{3} + y_{3} + \frac{1}{2} (x_{1}y_{2} - x_{2} y_{1}))$

multiplication by positive scalars

$s (x_{1},x_{2},x_{3}) = (sx_{1}, sx_{2}, s^{2}x_{3})$

and norm given by

$\| (x_{1}, x_{2}, x_{3}) \|^{2} = (x_{1})^{2} + (x_{2})^{2} + \mid x_{3} \mid$

Then we have $X \cdot Y = XY$, for any $X,Y \in G$ and $\| X\|_{0} = \|X\|$ for any $X \in G$.

Carnot groups are therefore just a noncommutative generalization of vector spaces, with the addition operation $+$ replaced by a noncommutative operation!

5. There are many groups with dilations which are not Carnot groups. For example endow any Lie group with a left invariant sub-riemannian structure and hop, this gives a norm group with dilations structure.

In such a group with dilations the “radial homogeneity” condition of Tao implies that the operation $x \cdot y$ is commutative! (see the references given in this previous post). Indeed, this radial homogeneity is equivalent with the following assertion: for any $s \in (0,1)$ and any $x, y \in G$

$x s( x^{-1} ) = (1-s)x$

which is called elsewhere “barycentric condition”. This condition is false in any noncommutative Carnot group! What it is true is the following: let, in a Carnot group, $x$ be any solution of the equation

$x s( x^{-1} ) = y$

for given $y \in G$ and $s \in (0,1)$. Then

$x = \sum_{k=0}^{\infty} (s^{k}) y$ ,

(so the solution is unique) where the sum is taken with respect to the group operation (noncommutative series).

Problem of the noncommutative BCH formula: In a normed group with dilations, express the group operation $xy$ as a noncommutative series, by using instead of “$+$” the operation “$\cdot$” and by using a definition of the “noncommutative Lie bracket” in the same spirit as (2), that is something related to the asymptotic behaviour of the “approximate bracket”

(3)         $[x,y]_{s} = (s^{-1}) ( x^{-1} \cdot (x *_{s} y) \cdot y^{-1} )$.

Notice that there is NO CHANCE to have a limit like the one in (2), so the problem seems hard also from this point of view.

Also notice that if $G$ is a Carnot group then

$[x,y]_{s} = e$ (that is like it is equal to $o$, remember)

which is normal, if we think about $G$ as being a kind of noncommutative vector space, even of $G$ may be not commutative.

So this noncommutative Lie bracket is not about commutators!