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Today we will define some basic linear algebra over the absolute fields $\mathbb{F}_{1^n} $ following the Kapranov-Smirnov document. Recall from last time that $\mathbb{F}_{1^n} = \mu_n^{\bullet} $ and that a d-dimensional vectorspace over this field is a pointed set $V^{\bullet} $ where $V $ is a free $\mu_n $-set consisting of n.d elements. Note that in absolute linear algebra we are not allowed to have addition of vectors and have to define everything in terms of scalar multiplication (or if you want, the $\mu_n $-action). In the hope of keeping you awake, we will include an F-un interpretation of the power residue symbol.

Direct sums of vectorspaces are defined via $V^{\bullet} \oplus W^{\bullet} = (V \bigsqcup W)^{\bullet} $, that is, correspond to the disjoint union of free $\mu_n $-sets. Consequently we have that $dim(V^{\bullet} \oplus W^{\bullet}) = dim(V^{\bullet}) + dim(W^{\bullet}) $.

For tensor-product we start with $V^{\bullet} \times W^{\bullet} = (V \times W)^{\bullet} $ the vectorspace cooresponding to the Cartesian product of free $\mu_n $-sets. If the dimensions of $V^{\bullet} $ and $W^{\bullet} $ are respectively d and e, then $V \times W $ consists of n.d.n.e elements, so is of dimension n.d.e. In order to have a sensible notion of tensor-products we have to eliminate the n-factor. We do this by identifying $~(x,y) $ with $(\epsilon_n x, \epsilon^{-1} y) $ and call the corresponding vectorspace $V^{\bullet} \otimes W^{\bullet} $. If we denote the image of $~(x,y) $ by $x \otimes w $ then the identification merely says we can pull the $\mu_n $-action through the tensor-sign, as we’d like to do. With this definition we do indeed have that $dim(V^{\bullet} \otimes W^{\bullet}) = dim(V^{\bullet}) dim(W^{\bullet}) $.

Recall that any linear automorphism $A $ of an $\mathbb{F}_{1^n} $ vectorspace $V^{\bullet} $ with basis ${ b_1,\ldots,b_d } $ (representants of the different $\mu_n $-orbits) is of the form $A(b_i) = \epsilon_n^{k_i} b_{\sigma(i)} $ for some powers of the primitive n-th root of unity $\epsilon_n $ and some permutation $\sigma \in S_d $. We define the determinant $det(A) = \prod_{i=1}^d \epsilon_n^{k_i} $. One verifies that the determinant is multiplicative and independent of the choice of basis.

For example, scalar-multiplication by $\epsilon_n $ gives an automorphism on any $d $-dimensional $\mathbb{F}_{1^n} $-vectorspace $V^{\bullet} $ and the corresponding determinant clearly equals $det = \epsilon_n^d $. That is, the det-functor remembers the dimension modulo n. These mod-n features are a recurrent theme in absolute linear algebra. Another example, which will become relevant when we come to reciprocity laws :

Take $n=2 $. Then, a $\mathbb{F}_{1^2} $ vectorspace $V^{\bullet} $ of dimension d is a set consisting of 2d elements $V $ equipped with a free involution. Any linear automorphism $A~:~V^{\bullet} \rightarrow V^{\bullet} $ is represented by a $d \times d $ matrix having one nonzero entry in every row and column being equal to +1 or -1. Hence, the determinant $det(A) \in \{ +1,-1 \} $.

On the other hand, by definition, the linear automorphism $A $ determines a permutation $\sigma_A \in S_{2d} $ on the 2d non-zero elements of $V^{\bullet} $. The connection between these two interpretations is that $det(A) = sgn(\sigma_A) $ the determinant gives the sign of the permutation!

For a prime power $q=p^k $ with $q \equiv 1~mod(n) $, we have seen that the roots of unity $\mu_n \subset \mathbb{F}_q^* $ and hence that $\mathbb{F}_q $ is a vectorspace over $\mathbb{F}_{1^n} $. For any field-unit $a \in \mathbb{F}_q^* $ we have the power residue symbol

$\begin{pmatrix} a \\ \mathbb{F}_q \end{pmatrix}_n = a^{\frac{q-1}{n}} \in \mu_n $

On the other hand, multiplication by $a $ is a linear automorphism on the $\mathbb{F}_{1^n} $-vectorspace $\mathbb{F}_q $ and hence we can look at its F-un determinant $det(a \times) $. The F-un interpretation of a classical lemma by Gauss asserts that the power residue symbol equals $det(a \times) $.

An $\mathbb{F}_{1^n} $-subspace $W^{\bullet} $ of a vectorspace $V^{\bullet} $ is a subset $W \subset V $ consisting of full $\mu_n $-orbits. Normally, in defining a quotient space we would say that two V-vectors are equivalent when their difference belongs to W and take equivalence classes. However, in absolute linear algebra we are not allowed to take linear combinations of vectors…

The only way out is to define $~(V/W)^{\bullet} $ to correspond to the free $\mu_n $-set $~(V/W) $ obtained by identifying all elements of W with the zero-element in $V^{\bullet} $. But… this will screw-up things if we want to interpret $\mathbb{F}_q $-vectorspaces as $\mathbb{F}_{1^n} $-spaces whenever $q \equiv 1~mod(n) $.

For this reason, Kapranov and Smirnov invent the notion of an equivalence $f~:~X^{\bullet} \rightarrow Y^{\bullet} $ between $\mathbb{F}_{1^n} $-spaces to be a linear map (note that this means a set-theoretic map $X \rightarrow Y^{\bullet} $ such that the invers image of 0 consists of full $\mu_n $-orbits and is a $\mu_n $-map elsewhere) satisfying the properties that $f^{-1}(0) = 0 $ and for every element $y \in Y $ we have that the number of pre-images $f^{-1}(y) $ is congruent to 1 modulo n. Observe that under an equivalence $f~:~X^{\bullet} \rightarrow Y^{\bullet} $ we have that $dim(X^{\bullet}) \equiv dim(Y^{\bullet})~mod(n) $.

This then allows us to define an exact sequence of $\mathbb{F}_{1^n} $-vectorspaces to be

with $\alpha $ a set-theoretic inclusion, the composition $\beta \circ \alpha $ to be the zero-map and with the additional assumption that the map induced by $\beta $

$~(V/V_1)^{\bullet} \rightarrow V_2^{\bullet} $

is an equivalence. For an exact sequence of spaces as above we have the congruence relation on their dimensions $dim(V_1)+dim(V_2) \equiv dim(V)~mod(n) $.

More importantly, if as before $q \equiv 1~mod(n) $ and we use the embedding $\mu_n \subset \mathbb{F}_q^* $ to turn usual $\mathbb{F}_q $-vectorspaces into absolute $\mathbb{F}_{1^n} $-spaces, then an ordinary exact sequence of $\mathbb{F}_q $-vectorspaces remains exact in the above definition.

All esoteric subjects have their own secret (sacred) texts. If you opened the Da Vinci Code (or even better, the original The Holy blood and the Holy grail) you will known about a mysterious collection of documents, known as the “Dossiers secrets“, deposited in the Bibliothèque nationale de France on 27 April 1967, which is rumoured to contain the mysteries of the Priory of Sion, a secret society founded in the middle ages and still active today…

The followers of F-un, for $\mathbb{F}_1 $ the field of one element, have their own collection of semi-secret texts, surrounded by whispers, of which they try to decode every single line in search of enlightenment. Fortunately, you do not have to search the shelves of the Bibliotheque National in Paris, but the depths of the internet to find them as huge, bandwidth-unfriendly, scanned documents.

The first are the lecture notes “Lectures on zeta functions and motives” by Yuri I. Manin of a course given in 1991.

One can download a scanned version of the paper from the homepage of Katia Consani as a huge 23.1 Mb file. Of F-un relevance is the first section “Absolute Motives?” in which

“…we describe a highly speculative picture of analogies between arithmetics over $\mathbb{F}_q $ and over $\mathbb{Z} $, cast in the language reminiscent of Grothendieck’s motives. We postulate the existence of a category with tensor product $\times $ whose objects correspond not only to the divisors of the Hasse-Weil zeta functions of schemes over $\mathbb{Z} $, but also to Kurokawa’s tensor divisors. This neatly leads to teh introduction of an “absolute Tate motive” $\mathbb{T} $, whose zeta function is $\frac{s-1}{2\pi} $, and whose zeroth power is “the absolute point” which is teh base for Kurokawa’s direct products. We add some speculations about the role of $\mathbb{T} $ in the “algebraic geometry over a one-element field”, and in clarifying the structure of the gamma factors at infinity.” (loc.cit. p 1-2)

I’d welcome links to material explaining this section to people knowing no motives.

The second one is the unpublished paper “Cohomology determinants and reciprocity laws : number field case” by Mikhail Kapranov and A. Smirnov.

This paper features in blog-posts at the Arcadian Functor, in John Baez’ Weekly Finds and in yesterday’s post at Noncommutative Geometry.

You can download every single page (of 15) as a separate file from here. But, in order to help spreading the Fun-gospel, I’ve made these scans into a single PDF-file which you can download as a 2.6 Mb PDF. In the introduction they say :

“First of all, it is an old idea to interpret combinatorics of finite sets as the $q \rightarrow 1 $ limit of linear algebra over the finite field $\mathbb{F}_q $. This had lead to frequent consideration of the folklore object $\mathbb{F}_1 $, the “field with one element”, whose vector spaces are just sets. One can postulate, of course, that $\mathbf{spec}(\mathbb{F}_1) $ is the absolute point, but the real problem is to develop non-trivial consequences of this point of view.”

They manage to deduce higher reciprocity laws in class field theory within the theory of $\mathbb{F}_1 $ and its field extensions $\mathbb{F}_{1^n} $. But first, let us explain how they define linear algebra over these absolute fields.

Here is a first principle : in doing linear algebra over these fields, there is no additive structure but only scalar multiplication by field elements. So, what are vector spaces over the field with one element? Well, as scalar multiplication with 1 is just the identity map, we have that a vector space is just a set. Linear maps are just set-maps and in particular, a linear isomorphism of a vector space onto itself is a permutation of the set. That is, linear algebra over $\mathbb{F}_1 $ is the same as combinatorics of (finite) sets.

A vector space over $\mathbb{F}_1 $ is just a set; the dimension of such a vector space is the cardinality of the set. The general linear group $GL_n(\mathbb{F}_1) $ is the symmetric group $S_n $, the identification via permutation matrices (having exactly one 1 in every row and column)

Some people prefer to view an $\mathbb{F}_1 $ vector space as a pointed set, the special element being the ‘origin’ $0 $ but as $\mathbb{F}_1 $ doesnt have a zero, there is also no zero-vector. Still, in later applications (such as defining exact sequences and quotient spaces) it is helpful to have an origin. So, let us denote for any set $S $ by $S^{\bullet} = S \cup { 0 } $. Clearly, linear maps between such ‘extended’ spaces must be maps of pointed sets, that is, sending $0 \rightarrow 0 $.

The field with one element $\mathbb{F}_1 $ has a field extension of degree n for any natural number n which we denote by $\mathbb{F}_{1^n} $ and using the above notation we will define this field as :

$\mathbb{F}_{1^n} = \mu_n^{\bullet} $ with $\mu_n $ the group of all n-th roots of unity. Note that if we choose a primitive n-th root $\epsilon_n $, then $\mu_n \simeq C_n $ is the cyclic group of order n.

Now what is a vector space over $\mathbb{F}_{1^n} $? Recall that we only demand units of the field to act by scalar multiplication, so each ‘vector’ $\vec{v} $ determines an n-set of linear dependent vectors $\epsilon_n^i \vec{v} $. In other words, any $\mathbb{F}_{1^n} $-vector space is of the form $V^{\bullet} $ with $V $ a set of which the group $\mu_n $ acts freely. Hence, $V $ has $N=d.n $ elements and there are exactly $d $ orbits for the action of $\mu_n $ by scalar multiplication. We call $d $ the dimension of the vectorspace and a basis consists in choosing one representant for every orbits. That is, $~B = { b_1,\ldots,b_d } $ is a basis if (and only if) $V = { \epsilon_n^j b_i~:~1 \leq i \leq d, 1 \leq j \leq n } $.

So, vectorspaces are free $\mu_n $-sets and hence linear maps $V^{\bullet} \rightarrow W^{\bullet} $ is a $\mu_n $-map $V \rightarrow W $. In particular, a linear isomorphism of $V $, that is an element of $GL_d(\mathbb{F}_{1^n}) $ is a $\mu_n $ bijection sending any basis element $b_i \rightarrow \epsilon_n^{j(i)} b_{\sigma(i)} $ for a permutation $\sigma \in S_d $.

An $\mathbb{F}_{1^n} $-vectorspace $V^{\bullet} $ is a free $\mu_n $-set $V $ of $N=n.d $ elements. The dimension $dim_{\mathbb{F}_{1^n}}(V^{\bullet}) = d $ and the general linear group $GL_d(\mathbb{F}_{1^n}) $ is the wreath product of $S_d $ with $\mu_n^{\times d} $, the identification as matrices with exactly one non-zero entry (being an n-th root of unity) in every row and every column.

This may appear as a rather sterile theory, so let us give an extremely important example, which will lead us to our second principle for developing absolute linear algebra.

Let $q=p^k $ be a prime power and let $\mathbb{F}_q $ be the finite field with $q $ elements. Assume that $q \cong 1~mod(n) $. It is well known that the group of units $\mathbb{F}_q^{\ast} $ is cyclic of order $q-1 $ so by the assumption we can identify $\mu_n $ with a subgroup of $\mathbb{F}_q^{\ast} $.

Then, $\mathbb{F}_q = (\mathbb{F}_q^{\ast})^{\bullet} $ is an $\mathbb{F}_{1^n} $-vectorspace of dimension $d=\frac{q-1}{n} $. In other words, $\mathbb{F}_q $ is an $\mathbb{F}_{1^n} $-algebra. But then, any ordinary $\mathbb{F}_q $-vectorspace of dimension $e $ becomes (via restriction of scalars) an $\mathbb{F}_{1^n} $-vector space of dimension $\frac{e(q-1)}{n} $.

Next time we will introduce more linear algebra definitions (including determinants, exact sequences, direct sums and tensor products) in the realm the absolute fields $\mathbb{F}_{1^n} $ and remarkt that we have to alter the known definitions as we can only use the scalar-multiplication. To guide us, we have the second principle : all traditional results of linear algebra over $\mathbb{F}_q $ must be recovered from the new definitions under the vector-space identification $\mathbb{F}_q = (\mathbb{F}_q^{\ast})^{\bullet} = \mathbb{F}_{1^n} $ when $n=q-1 $. (to be continued)

At the
noncommutative algebra program in MSRI 1999/2000, Mikhail Kapranov gave
an intriguing talk Noncommutative neighborhoods and noncommutative Fourier transform
and over the years I’ve watched the video of this talk a number
of times. The first part of the talk is about his work on Noncommutative geometry
based on commutator expansions and as I’ve once worked through it
this part didn’t present problems. On the other hand, I’ve never
understood much from the second part of the talk which claims to relate
these noncommutative formal neighborhoods to _noncommutative Fourier
transforms_. The string coffee table has a post Kapranov
and Getzler on Higher Stuff linking to two recent talks by Kapranov
on noncommutative Fourier transforms at the Streetfest. Marni
Sheppeard made handwritten notes available. I definitely should find the time
to get through them and have another go at the Kapranov-video…

Now that the preparation for my undergraduate courses in the first semester is more or less finished, I can begin to think about the courses I’ll give this year in the master class
non-commutative geometry. For a change I’d like to introduce the main ideas and concepts by a very concrete example : Ginzburg’s coadjoint-orbit result for the Calogero-Moser space and its
relation to the classification of one-sided ideals in the first Weyl algebra. Not only will this example give me the opportunity to say things about formally smooth algebras, non-commutative
differential forms and even non-commutative symplectic geometry, but it also involves what some people prefer to call _non-commutative algebraic geometry_ (that is the study of graded Noetherian
rings having excellent homological properties) via the projective space associated to the homogenized Weyl algebra. Besides, I have some affinity with this example.

A long time ago I introduced
the moduli spaces for one-sided ideals in the Weyl algebra in Moduli spaces for right ideals of the Weyl algebra and when I was printing a _very_ preliminary version of Ginzburg’s paper
Non-commutative Symplectic Geometry, Quiver varieties, and Operads (probably because he send a preview to Yuri Berest and I was in contact with him at the time about the moduli spaces) the
idea hit me at the printer that the right way to look at the propblem was to consider the quiver

$\xymatrix{\vtx{} \ar@/^/[rr]^a & & \vtx{} \ar@(u,ur)^x \ar@(d,dr)_y \ar@/^/[ll]^b} $

which eventually led to my paper together with Raf Bocklandt Necklace Lie algebras and noncommutative symplectic geometry.

Apart from this papers I would like to explain the following
papers by illustrating them on the above example : Michail Kapranov Noncommutative geometry based on commutator expansions Maxim Kontsevich and Alex Rosenberg Noncommutative smooth
spaces Yuri Berest and George Wilson Automorphisms and Ideals of the Weyl Algebra Yuri Berest and George Wilson Ideal Classes of the Weyl Algebra and Noncommutative Projective
Geometry Travis Schedler A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver and of course the seminal paper by Joachim Cuntz and Daniel Quillen on
quasi-free algebras and their non-commutative differential forms which, unfortunately, in not available online.

I plan to write a series of posts here on all this material but I will be very
happy to get side-tracked by any comments you might have. So please, if you are interested in any of this and want to have more information or explanation do not hesitate to post a comment (only
your name and email is required to do so, you do not have to register and you can even put some latex-code in your post but such a posting will first have to viewed by me to avoid cluttering of
nonsense GIFs in my directories).