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Tag: Quillen

path algebras

Published September 3, 2004 by lieven

The previous post can be found [here][1].
Pierre Gabriel invented a lot of new notation (see his book [Representations of finite dimensional algebras][2] for a rather extreme case) and is responsible for calling a directed graph a quiver. For example,

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

is a quiver. Note than it is allowed to have multiple arrows between vertices, as well as loops in vertices. For us it will be important that a quiver $Q $ depicts how to compute in a certain non-commutative algebra : the path algebra $\mathbb{C} Q $. If the quiver has $k $ vertices and $l $ arrows (including loops) then the path algebra $\mathbb{C} Q $ is a subalgebra of the full $k \times k $ matrix-algebra over the free algebra in $l $ non-commuting variables

$\mathbb{C} Q \subset M_k(\mathbb{C} \langle x_1,\ldots,x_l \rangle) $

Under this map, a vertex $v_i $ is mapped to the basis $i $-th diagonal matrix-idempotent and an arrow

$\xymatrix{\vtx{v_i} \ar[rr]^{x_a} & & \vtx{v_j}} $

is mapped to the matrix having all its entries zero except the $(j,i) $-entry which is equal to $x_a $. That is, in our main example

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

the corresponding path algebra is the subalgebra of $M_2(\mathbb{C} \langle a,b,x,y \rangle) $ generated by the matrices

$e \mapsto \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} $ $ f \mapsto \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} $

$a \mapsto \begin{bmatrix} 0 & 0 \\ a & 0 \end{bmatrix} $ $b \mapsto \begin{bmatrix} 0 & b \\ 0 & 0 \end{bmatrix} $

$x \mapsto \begin{bmatrix} 0 & 0 \\ 0 & x \end{bmatrix} $ $y \mapsto \begin{bmatrix} 0 & 0 \\ 0 & y \end{bmatrix} $

The name \’path algebra\’ comes from the fact that the subspace of $\mathbb{C} Q $ at the $(j,i) $-place is the vectorspace spanned by all paths in the quiver starting at vertex $v_i $ and ending in vertex $v_j $. For an easier and concrete example of a path algebra. consider the quiver

$\xymatrix{\vtx{e} \ar[rr]^a & & \vtx{f} \ar@(ur,dr)^x} $

and verify that in this case, the path algebra is just

$\mathbb{C} Q = \begin{bmatrix} \mathbb{C} & 0 \\ \mathbb{C}[x]a & \mathbb{C}[x] \end{bmatrix} $

Observe that we write and read paths in the quiver from right to left. The reason for this strange convention is that later we will be interested in left-modules rather than right-modules. Right-minder people can go for the more natural left to right convention for writing paths.
Why are path algebras of quivers of interest in non-commutative geometry? Well, to begin they are examples of _formally smooth algebras_ (some say _quasi-free algebras_, I just call them _qurves_). These algebras were introduced and studied by Joachim Cuntz and Daniel Quillen and they are precisely the algebras allowing a good theory of non-commutative differential forms.
So you should think of formally smooth algebras as being non-commutative manifolds and under this analogy path algebras of quivers correspond to _affine spaces_. That is, one expects path algebras of quivers to turn up in two instances : (1) given a non-commutative manifold (aka formally smooth algebra) it must be \’embedded\’ in some non-commutative affine space (aka path algebra of a quiver) and (2) given a non-commutative manifold, the \’tangent spaces\’ should be determined by path algebras of quivers.
The first fact is easy enough to prove, every affine $\mathbb{C} $-algebra is an epimorphic image of a free algebra in say $l $ generators, which is just the path algebra of the _bouquet quiver_ having $l $ loops

$\xymatrix{\vtx{} \ar@(dl,l)^{x_1} \ar@(l,ul)^{x_2} \ar@(ur,r)^{x_i} \ar@(r,dr)^{x_l}} $

The second statement requires more work. For a first attempt to clarify this you can consult my preprint [Qurves and quivers][3] but I\’ll come back to this in another post. For now, just take my word for it : if formally smooth algebras are the non-commutative analogon of manifolds then path algebras of quivers are the non-commutative version of affine spaces!

[1]: https://lievenlb.local/index.php?p=71
[2]: http://www.booxtra.de/verteiler.asp?site=artikel.asp&wea=1070000&sh=homehome&artikelnummer=000000689724
[3]: http://www.arxiv.org/abs/math.RA/0406618

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nog course outline

Published September 2, 2004 by lieven

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).

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the one quiver for GL(2,Z)

Published August 26, 2004 by lieven

Before the vacation I finished a rewrite of the One quiver to rule them
all note. The main point of that note was to associate to any qurve
$A$ (formerly known as a quasi-free algebra in the terminology of
Cuntz-Quillen or a formally smooth algebra in the terminology of
Kontsevich-Rosenberg) a quiver $Q(A)$ and a dimension vector $\alpha_A$
such that $A$ is etale isomorphic (in a yet to be defined
non-commutative etale toplogy) to a ring Morita equivalent to the path
algebra $lQ(A)$ where the Morita setting is determined by the dimension
vector $\alpha_A$. These “one-quiver settings” are easy to
work out for a group algebra $lG$ if $G$ is the amalgamated free product
of finite groups $G = H_1 \bigstar_H H_2$.

Here is how to do
this : construct a bipartite quiver with the left vertices corresponding
to the irreducible representations of $H_1$, say ${ S_1, .. ,S_k }$ of
dimensions $(d_1, .. ,d_k)$ and the right vertices corresponding to the
irreducible representations of $H_2$, ${ T_1, .. ,T_l }$ of dimensions
$(e_1, .. ,e_l)$. The number of arrows from the $i$-th left vertex to
the $j$-th right vertex is given by the dimension of $Hom_H(S_i,T_j)$
This is the quiver I call the Zariski quiver for $G$ as the finite
dimensional $G$-representations correspond to $\theta$-semistable
representations of this quiver for the stability structure $\theta=(d_1,
.. ,d_k ; -e_1, .. ,-e_l)$. The one-quiver $Q(G)$ has vertices
corresponding to the minimal $\theta$-stable dimension vectors (say
$\alpha,\beta, .. $of the Zariski quiver and with the number of arrows
between two such vertices determined by $\delta_{\alpha
\beta}-\chi(\alpha,\beta)$ where $\chi$ is the Euler form of the Zariski
quiver. In the old note I've included the example of the projective
modular group $PSL_2(Z) = Z_2 \bigstar Z_3$ (which can easily be
generalized to the modular group $SL_2(Z) = Z_4 \bigstar_{Z_2} Z_6$)
which turns out to be the double of the extended Dynkin quiver
$\tilde{A_5}$. In the rewrite I've also included an example of a
congruence subgroup $\Gamma_0(2) = Z_4 \bigstar_{Z_2}^{HNN}$ which is an
HNN-extension. These are somehow the classical examples of interesting
amalgamated (HNN) groups and one would like to have plenty of other
interesting examples. Yesterday I read a paper by Karen Vogtmann called

Automorphisms of free groups and outer space in which I encountered
an amalgamated product decomposition for $GL_2(Z) = D_8 \bigstar_{Z_2
\times Z_2} (S_3 \times Z_2)$where $D_8$ is the diheder group of 8
elements. When I got back from vacation I found a reference to this
result in my mail-box from Warren Dicks. Theorem 23.1, p. 82, in Heiner
Zieschang, Finite Groups of Mapping Classes of Surfaces, LNM 875,
Springer, Berlin, 1981.

I worked out the one-quiver and it has
the somewhat strange form depicted above. It is perfectly possible that
I made mistakes so if you find another result, please let me know.

added material (febr 2007) : mistakes were made and
the correct one quiver can be found elsewhere on this blog.

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