Archive for April, 2009

0-stable Modules Part 2

April 25, 2009

Fix a directed graph (I,\Omega) and its double (I,H).  Fix I-graded vector spaces V,W.

Let h_\bullet=(h_n,... h_1) be a path in (I,H).  That is, h_i\in H and s(h_{k+1})=e(h_k) for 1\le k\le n-1.  The length of h_\bullet is n.  Define s(h_\bullet)=s(h_1) and e(h_\bullet)=e(h_n).

P(H) will denote the set of all paths,  P(i,H) will denote the set of paths begining at i, and P(H,j) will denote the paths ending at j.  In practice we will actually these with finite sets depending on the dimension of V — I’ll address this in a moment.

Let (B,a,b) be a module as defined in the previous post.  B_h:V_{s(h)}\rightarrow V_{e(h)} refers to the component of B associated to a particular edge h and B_{h_\bullet}:V_{s(h_\bullet)}\rightarrow V_{e(h_\bullet)} refers a composition of such maps.

Then we may alternatively define zero-stable as follows: (B,a,b) is zero-stable if for all i\in I and every y \in V_i

  1. there exists a path h_\bullet\in P(i,H) such that b_{e(h_\bullet)}B_{h_\bullet}(y)\neq 0 and
  2. there exists a vector x\in\bigoplus_{h_\bullet\in P(H,i)} W_{s(h_\bullet)} such that \sum B_{h_\bullet}a_{s(h_\bullet)}(x_{s(h_\bullet)})=y.

The proof that this is equivelant to the previous definition is straightforward and also demonstrates that we need only consider paths that visit each vertex i no more than \dim(V_i) times.

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0-stable Modules

April 24, 2009

For Nakajima’s analogue of MV-cycles we need to understand what it takes for a “module” to be 0-stable. Let me review first what that means.

Fix a finite set I={1 .. r} and a directed graph (I,\Omega). Let H=\Omega\sqcup\bar\Omega. For two I-graded vector spaces V and W we define
E(V,W)=\bigoplus_{h\in H} Hom(V_{s(h)},W_{e(h)}
L(V,W)=\bigoplus_{i\in I} Hom(V_i,W_i)
M(V,W)=E(V,V)\oplus L(W,V) \oplus L(V,W).
(where s and e denote the start and end vertices of an edge.)
We have a function
\epsilon:E(V,V)\rightarrow E(V,V)
which multiplies by -1 components corresponding to h\in\bar\Omega and fixes components corresponding to h\in\Omega. We have a composition
L(V,W)\times L(W,X) \rightarrow L(V,X)
which is straightforward. We have a composition
E(V,W)\times E(W,X) \rightarrow L(V,X)
where (CB)_i=\sum_{e(h)=i} C_hB_{\bar h}. And we have a “moment map vanishing at the origin”
\mu:M(V,W)\rightarrow L(V,V)
where \mu(B,a,b)=(\epsilon B)B+ab.

From now on we fix V and W.

A point (B,a,b)\in\mu^{-1}(0) is called a “module”.

A “sub-module” of (B,a,b) is a B-invariant I-graded vector space V'\subset V which either contains Im(a) or is contained in Ker(b).

A module (B,a,b) is said to be “0-stable” if the only sub-modules are 0 and V.

**EDIT: I should say (B,a,b) is zero-stable if the only sub-module containing Im(a) is V and the only submodule contained in Ker(b) is 0.

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