coalgebras and non-geometry 2

By lieven

Last time we have seen that the coalgebra of distributions of an affine smooth variety is the direct sum (over all points) of the dual to the etale local algebras which are all of the form $\mathbb{C}[[ x_1,\hdots,x_d ]]$ where is the dimension of the variety. Generalizing this to non-commutative manifolds, the first questions are : “What is the analogon of the power-series algebra?” and do all ‘points’ of our non-commutative manifold do have such local algebras? Surely, we no longer expect the variables to commute, so a non-commutative version of the power series algebra should be $\mathbb{C} \langle \langle x_1,\hdots,x_d \rangle \rangle$ , the ring of formal power series in non-commuting variables. However, there is still another way to add non-commutativity and that is to go from an algebra to matrices over the algebra. So, in all we would expect to be our local algebras at points of our non-commutative manifold to be isomorphic to $M_n(\mathbb{C} \langle \langle x_1,\hdots,x_d \rangle \rangle)$ As to the second question : _qurves (that is, the coordinate rings of non-commutative manifolds) do have such algebras as local rings provided we take as the ‘points’ of the non-commutative variety the set of all simple finite dimensional representations of the qurve. This is a consequence of the tubular neighborhood theorem due to Cuntz and Quillen. In more details : If A is a qurve, then a simple -dimensional representation corresponds to an epimorphism $\pi~:~A \rightarrow S = M_n(\mathbb{C})$ and if we take $\mathfrak{m}=Ker(\pi)$ , then $M=\mathfrak{m}/\mathfrak{m}^2$ is an -bimodule and the $\mathfrak{m}$ -adic completion of A is isomorphic to the completed tensor-algebra $\hat{T}_S(M) \simeq M_n(\mathbb{C} \langle \langle x_1,\hdots,x_d \rangle \rangle)$ In contrast with the commutative case however where the dimension remains constant over all points, here the numbers n and d can change from simple to simple. For n this is clear as it gives the dimension of the simple representation, but also d changes (it is the local dimension of the variety classifying simple representations of the same dimension). Here an easy example : Consider the skew group algebra $A = \mathbb{C}[x] \star C_2$ with the action given by sending $x \mapsto -x$ . Then A is a qurve and its center is $\mathbb{C}[y]$ with y=x^2 . Over any point $y \not= 0$ there is a unique simple 2-dimensional representation of A giving the local algebra $M_2(\mathbb{C}[[y]])$ . If y=0 the situation is more complicated as the local structure of A is given by the algebra $\begin{bmatrix} \mathbb{C}[[y]] & \mathbb{C}[[y]] \\ (y) & \mathbb{C}[[y]] \end{bmatrix}$ So, over this point there are precisely 2 one-dimensional simple representations corresponding to the maximal ideals $\mathfrak{m}_1 = \begin{bmatrix} (y) & \mathbb{C}[[y]] \\ (y) & \mathbb{C}[[y]] \end{bmatrix}~\qquad \text{and}~\qquad \mathfrak{m}_2 = \begin{bmatrix} \mathbb{C}[[y]] & \mathbb{C}[[y]] \\ (y) & (y) \end{bmatrix}$ and both ideals are idempotent, that is $\mathfrak{m}_i^2 = \mathfrak{m}_i$ whence the corresponding bimodule M_i =
0 so the local algebra in either of these two points is just $\mathbb{C}$ . Ok, so the comleted local algebra at each point is of the form $M_n(\mathbb{C}\langle \langle x_1,\hdots,x_d \rangle \rangle)$ , but what is the corresponding dual coalgebra. Well, $\mathbb{C} \langle \langle x_1,\hdots,x_d \rangle \rangle$ is the algebra dual to the _cofree coalgebra on $V = \mathbb{C} x_1 + \hdots + \mathbb{C}x_d$ . As a vectorspace this is the tensor-algebra $T(V) = \mathbb{C} \langle x_1,\hdots,x_d \rangle$ with the coalgebra structure induced by the bialgebra structure defined by taking all varaibales to be primitives, that is $\Delta(x_i) = x_i \otimes 1 + 1 \otimes x_i$ . That is, the coproduct on a monomial gives all different expressions $m_1 \otimes m_2$ such that m_1m_2 = m . For example, $\Delta(x_1x_2) = x_1x_2 \otimes 1 + x_1 \otimes x_2 + 1 \otimes x_1x_2$ . On the other hand, the dual coalgebra of $M_n(\mathbb{C})$ is the matrix coalgebra which is the n^2 -dimensional vectorspace $\mathbb{C}e_{11} + \hdots + \mathbb{C}e_{nn}$ with comultiplication $\Delta(e_{ij}) = \sum_k e_{ik} \otimes e_{kj}$ The coalgebra corresponding to the local algebra $M_n(\mathbb{C}\langle \langle x_1,\hdots,x_d \rangle \rangle)$ is then the tensor-coalgebra of the matrix coalgebra and the cofree coalgebra. Having obtained the coalgebra at each point (=simple representation) of our noncommutative manifold one might think that the _coalgebra of non-commutative distributions should be the direct sum of all this coalgebras, summed over all points, as in the commutative case. But then we would forget about a major difference between the commutative and the non-commutative world : distinct simples can have non-trivial extensions! The mental picture one might have about simples having non-trivial extensions is that these points lie ‘infinitesimally close’ together. In the $\mathbb{C}[x] \star C_2$ example above, the two one-dimensional simples have non-trivial extensions so they should be thought of as a cluster of two infinitesimally close points corresponding to the point y=0 (that is, this commutative points splits into two non-commutative points). Btw. this is the reason why non-commutative algebras can be used to resolve commutative singularities (excessive tangents can be split over several non-commutative points). While this is still pretty harmless when the algebra is finite over its center (as in the above example where only the two one-dimensionals have extensions), the situation becomes weird over general qurves as ‘usually’ distinct simples have non-trivial extensions. For example, for the free algebra $\mathbb{C}\langle x,y \rangle$ this is true for all simples… So, if we want to continue using this image of points lying closely together this immediately means that non-commutative ‘affine’ manifolds behave like compact ones (in fact, it turns out to be pretty difficult to ‘glue’ together qurves into ‘bigger’ non-commutative manifolds, apart from the quiver examples of this old paper). So, how to bring this new information into our coalgebra of distributions? Well, let’s repeat the previous argument not with just one point but with a set of finitely many points. Then we have a _semi-simple algebra quotient $\pi~:~A \rightarrow S = M_{n_1}(\mathbb{C}) \oplus \hdots \oplus M_{n_k}(\mathb{C})$ and taking again $\mathfrak{m}=Ker(\pi)$ and $M=\mathfrak{m}/\mathfrak{m}^2$ , then is again an S-bimodule. Now, any S-bimodule can be encoded into a quiver Q on k points, the number of arrows from vertex i to vertex j being the number of components in M of the form $M_{n_i \times n_j}(\mathbb{C})$ . Again, it follows from the tubular neighborhood theorem that the $\mathfrak{m}$ -adic completion of A is isomorphic to the completion of an algebra Morita equivalent to the _path algebra $\mathbb{C} Q$ (being the tensor algebra T_S(M) ). As all the local algebras of the points are quotients of this quiver-like completion, on the coalgebra level our local coalgebras will be sub coalgebras of the coalgebra which is co-Morita equivalent (and believe it or not but coalgebraists have a name for this : _Takeuchi equivalence) to the quiver coalgebra which is the vectorspace of the path algebra $\mathbb{C} Q$ with multiplication induced by making all arrows from i to j skew-primitives, that is, $\Delta(a) = e_i \otimes a + a \otimes e_j$ where the e_i are group-likes corresponding to the vertices. If all of ths is a bit too much co to take in at once, I suggest the paper by Bill Chin A brief introduction to coalgebra representation theory. The _coalgebra of noncommutative distributions we are after at is now the union of all these Takeuchi-equivalent quiver coalgebras. In easy examples such as the $\mathbb{C}[x] \star C_2$ -example this coalgebra is still pretty small (the sum of the local coalgebras corresponding to the local algebras $M_2(\mathbb{C}[[x]])$ summed over all points $y \not= 0$ summed with the quiver coalgebra of the quiver $\xymatrix{\vtx{} \ar@/^/[rr] & & \vtx{} \ar@/^/[ll]}$ In general though this is a huge object and we would like to have a recipe to construct it from a manageable blue-print and that is what we will do next time.