CRYSTALS FROM PREPROJECTIVE ALGEBRAS OF MODULATED
GRAPHS
VINOTH NANDAKUMAR AND PETER TINGLEY
Abstract. Kashiwara and Saito give a geometric construction of the crystal B(−∞) for
a symmetric Kac-Moody algebra by using irreducible components of Lusztig’s quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. Here we
generalize their construction to symmetrizable Kac-Moody algebras by replacing Lusztig’s
preprojective algebra with a more general one due to Dlab and Ringel. This requires nontrivial field extensions (in non-symmetric cases), so we must work over fields which are
not algebraically closed. Our work also gives a non-trivial deformation of Kashiwara and
Saito’s realization even in symmetric type and working over C.
Contents
1. Introduction
1.1. Overview
1.2. Modulated graphs and preprojective algebras
1.3. Nilpotent Representation varieties
1.4. Summary of main result
1.5. Connections with the literature
1.6. Acknowledgements
2. Background
2.1. Symmetrizable Kac-Moody algebras
2.2. Crystals
2.3. Some topology
3. Representation varieties and crystals
3.1. Definition of the representation variety
3.2. Some important spaces.
3.3. Relations between components
3.4. Crystal operators
3.5. Reworded operators
3.6. Realization of B(−∞)
4. Examples
4.1. Continuing type C2
b2
4.2. Deformed construction over C for sl
References
2010 Mathematics Subject Classification. 17B37,16A64.
Key words and phrases. Crystal, pre-projective algebra, quiver variety, Kac-Moody algebra.
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VINOTH NANDAKUMAR AND PETER TINGLEY
1. Introduction
1.1. Overview. Fix a symmetrizable Kac-Moody algebra g. Kashiwara’s crystal B(−∞)
is a combinatorial object (a discrete set along with some partial-permutations f˜i ) that
encodes a lot of information about g and its integrable representations. It is usually defined
using the corresponding quantized universal enveloping algebra, but it can also be realized
by other methods. In symmetric type, Kashiwara and Saito [KS97] developed a very useful
geometric realization, where the underlying set consists of the collection of irreducible
components of some algebraic varieties.
The varieties used in Kashiwara and Saito’s construction are Lustzig’s nilpotent varieties
from [Lus91, §12]. They are the varieties of nilpotent representations of Lusztig’s preprojective algebra Λ acting on a fixed (graded) vector space. Lustzig’s preprojective algebra
is only defined in symmetric cases, which is why Kashiwara and Saito’s construction is
restricted to that generality.
However, even before Lusztig’s work, Dlab and Ringel [DR80] define the preprojective
algebra of a “modulated graph”. There is a natural way to associate a symmetrizable (but
not necessarily symmetric) Cartan datum to any modulated graph, and all symmetriable
Cartan matrices arise this way. If that Cartan datum happens to be symmetric then, by
making the appropriate choices in Dlab and Ringel’s construction, one recovers Lusztig’s
preprojective algebra (see [Rin98]).
Our main result is to generalize Kashiwara and Saito’s realization of B(−∞) by replacing
Lusztig’s preprojective algebra with Ringel and Dlab’s version. This gives a realization of
B(−∞) for any symmetrizable Kac-Moody algebra.
1.2. Modulated graphs and preprojective algebras. Modulated graphs (also sometimes called species) date back to work of Gabriel [Gab73]. The preprojective algebra
construction here is due to Dlab and Ringel [DR80].
Fix an undirected graph Γ, and denote the set of vertices by I and the set of edges by
E. Let A be the set of directed edges, which we will call arrows; so there are two arrows
in A for each edge in E. We do not allow edges connecting a vertex to itself, or multiple
edges. Denote the arrow from i to j by j ai ; since we do not allow multiple edges this does
not cause confusion.
A modulated graph M is a graph Γ along with a choice of a field F and:
• A choice of field Fi for each vertex i of the graph, such that Fi is a finite extension
of F, and ∩i Fi = F.
• For each arrow j ai , an (Fj , Fi ) bimodule j Mi such that the two actions of F ⊂ Fi , Fj
agree.
• For each arrow j ai , a non-degenerate Fi -bilinear form ji : i Mj ⊗Fj j Mi → Fi .
The tensor algebra TM is
M
(1)
TM =
k Mk−1
⊗Fk−1 · · · ⊗Fi3
i 2 Mi 2
⊗Fi2
i 2 Mi 1 ,
i1 i2 ···ik a path in Γ
with multiplication being tensor product if the end of one path agrees with the beginning
of the next, and 0 otherwise.
CRYSTALS AND MODULATED GRAPHS
3
For each arrow j ai , the bilinear form ji defines a canonical element rji in j Mi ⊗Fi i Mj ,
which can be taken to be
X
(2)
rji :=
vk ⊗ v k
k
for any pair of dual Fi bases {vk } ⊂ j Mi , {v k } ⊂ i Mj with respect to ji . It is well known
that this does not depend on the choice of dual bases. Furthermore, although the two Fj
actions on j Mi ⊗Fi i Mj need not agree, it is true that zri = ri z for all z ∈ Fj .
For each i ∈ I, define
X j
(3)
ri :=
ri
j:j ai ∈A
Definition 1.1. (see [DR80]) The preprojective algebra ΛM is the quotient of TM by
{ri }i∈I .
One can associate a symmetrizable (but not necessarily symmetric) Cartan matrix C =
(cij ) to such a modulated graph as follows:


2 if i = j
(4)
ci,j = − dimFi i Mj if there is an arrow from i to j


0
otherwise.
As in [DR80], ΛM is finite dimensional over F if and only if C is of finite type. If C
is symmetric then, letting Fi = C for all i and taking an obvious choice of bimodules
and bilinear forms, one recovers Lusztig’s preprojective algebra from [Lus91] (see [Rin98]).
However, even in this case, different choices of bilinear form give non-isomorphic algebras
(see [Rin98] or §4.2).
Example 1.2. Consider the modulated graph with F1 = R, F2 = C, 1 M2 = 2 M1 = C,
with the standard actions of R and C by multiplication. Use the bilinear forms:
(5)
21 : C ⊗C C → R
z ⊗ w → Re (zw)
12 : C ⊗R C → C
z ⊗ w → zw
The corresponding Cartan matrix is of type C2 . Consider the elements of the tensor algebra:
• e1 = 1 ∈ F1 and e2 = 1 ∈ F2 in degree 0.
• τ = 1 ∈ 2 M1 and τ̄ = 1 ∈ 1 M2 in degree 1.
then the relations defining the preprojective algebra Λ are
(6)
ττ = 0
and
τ τ − iτ τ i = 0.
As a vector space, the preprojective algebra decomposes as
(7)
Re1 ⊕ Ce2 ⊕ Cτ ⊕ Rτ ⊕ Rτ i ⊕ Cτ τ ⊕ Rτ iτ,
where the field shown in each case is the one that naturally acts on the left.
4
VINOTH NANDAKUMAR AND PETER TINGLEY
1.3. Nilpotent P
Representation varieties. There is a natural partition of the identity
e ∈ ΛM as e = i ei , where ei is the lazy
L path at node i. Given a representation of ΛM
on a vector space V , notice that V = i ei V , and in fact each ei V is naturally a left Fi
module. Given a dimension vector v = (vi )i∈I , fix a vi dimensional left Fi module Vi for
each i. Define the nilpotent representation variety Λ(ν) to be the variety of representations
of ΛM on V = ⊕i Vi such that ei V = Vi , the induced left Fi module structure on Vi agrees
with the original Fi module structure, and all sufficiently long paths act as 0. Note that,
up to isomorphism, Λ(ν) does not depend on any choice.
The representation variety Λ(ν) is contained in
M
HomFj (j Mi ⊗Fi Vi , Vj ),
(8)
(i,j)∈A
and is cut out by the polynomial equations stating that
• each ri acts as 0, and
P
• for all paths of length at least i dimFi Vi , every element in the right side of (1)
acts as 0.
This is the set of F points of an algebraic variety, so inherits the Zariski topology.
1.4. Summary of main result. We will show that the union over all v of the set of
irreducible components of Λ(v) realizes B(−∞), where the crystal operators are natural
analogues of those used by Kashiwara and Saito [KS97]. We also establish some related
results, such as the fact that Λ(v) is always pure dimensional, with each irreducible component being quasi-affine. To a large extent, Kashiwara and Saito’s original proof goes
through, although we have made a few modifications.
We then give a few examples of the new behavior our construction exhibits, and discuss
some potential applications. Interestingly, our construction is more general than Lusztig’s
even when the Cartan matrix is symmetric and all the Fi are chosen to be C. See §4.2.
1.5. Connections with the literature. There is already a well known way to study
B(−∞) in symmetrizable types by “folding” the quiver variety for a larger symmetric
Kac-Moody algebra (see [Sav05]). There the crystal for the symmetrizable Kac-Moody
algebra is the set of irreducible components of the quiver variety for the symmetric KacMoody algebra that are fixed set-wise by an automorphism related to an automorphism
of the Dynkin diagram. Due to that construction, most results that can be proven about
B(∞) using quiver varieties have been extended to symmetrizable type. However, with our
symmetrizable quiver varieties, the proofs of those results should be simplified, as many of
the symmetric type proofs should simply carry over without modification. We also feel it
aesthetically important to have a quiver variety in symmetrizable types that is actually a
representation variety for some algebra.
One could hope to see a direct relationship between our construction and the folding
construction, and in fact we believe this is possible: It seems that, by composing the
Diagram automorphism with a Galois automorphism of the field that has the same order,
the irreducible components that are fixed set-wise are in bijection with the irreducible
components of the set of actual fixed points, which is in turn isomorphic to our quiver
variety. Perhaps this will be the subject of a future work.
CRYSTALS AND MODULATED GRAPHS
5
This paper does not consider the natural question of where U − (g) itself can be realized
using our quiver varieties, in a way analogous to the symmetric case.
We also note that the current work can perhaps be generalized: Dlab and Ringel actually
allow division rings everywhere where we have used fields. We believe that one should be
able to construct representation varieties for these more general modulated graphs and
still realize B(−∞), but this involves some technicalities which we prefer to avoid for the
moment.
Finally, we note two recent papers of Geiss, Leclerc and Schroer [GLSa, GLSb] which
also address preprojective algebras in symmetrizable type. They take a different approach,
via quivers with relations, and so their constructions is quite different from ours. They do
not consider crystals, but do have a realization of U − (g).
1.6. Acknowledgements. The ideas that eventually led to this paper arose from a series
of discussions with Gordana Todorov, beginning at the Maurice Auslander lectures in 2010.
We thank Gordana for all her contributions, and the organizers of the Auslander lectures
for that wonderful event. We also thank George Lusztig and Pavel Etingof for interesting
comments in the early stages of this work. The second author was partially supported by
NSF grants DMS-0902649 and DMS-1265555.
2. Background
2.1. Symmetrizable Kac-Moody algebras. Let g be a symmetrizable Kac-Moody algebra with Cartan matrix C = (cij )i∈I , and let D = diag{di }i∈I be such that DC is symmetric, with the di relatively prime positive integers. Let P be the weight lattice of g and Q
the root lattice, and denote by {αi } be the simple roots. Let Q+ = spanZ≥0 {αi }i∈I . Recall
the usual (symmetric) bilinear form on Q defined by (αi , αj ) = di cij . We also have the form
(α ,α )
h·, ·i between the root lattice, and the co-root lattice, defined by hαi , αˇj i = 2 (αji ,αjj ) = cj,i .
2.2. Crystals. The following definitions and results are by now standard. Here we roughly
follow [Kas95] and [KS97, §3].
Definition 2.1. (see [Kas95, §7.2]) A combinatorial crystal is a set B along with
functions wt : B → P (where P is the weight lattice), and, for each i ∈ I, functions
εi , ϕi : B → Z ∪ {−∞} and ei , fi : B → B t {∅}, such that
(i)
(ii)
(iii)
(iv)
ϕi (b) = εi (b) + hwt(b), αi∨ i.
ei increases ϕi by 1, decreases εi by 1 and increases wt by αi .
fi b = b0 if and only if ei b0 = b.
If ϕi (b) = −∞, then ei b = fi b = ∅.
We often denote a combinatorial crystal simply by B, suppressing the other data.
Definition 2.2. A lowest weight combinatorial crystal is a combinatorial crystal with a
distinguished element b− (the lowest weight element) such that
(i) The element b− can be reached from any b ∈ B by applying a finite sequence of fi
for various i ∈ I.
(ii) For all b ∈ B and all i ∈ I, ϕi (b) = max{n : fin (b) 6= ∅}.
6
VINOTH NANDAKUMAR AND PETER TINGLEY
For lowest weight combinatorial crystals, the functions ϕi , εi and wt are determined from
the operators ei , fi and the weight of b− .
Here we are concerned with the “infinity crystal” B(−∞), which can be thought of as
the crystal for Uq− (g). We will use the following result essentially as our definition. It is
a rewording of [KS97, Proposition 3.2.3] designed to make the roles of the usual crystal
operators and the ∗-crystal operators more symmetric; see [TW] for this exact statement.
Proposition 2.3. Fix a set B with operators ei , fi , e∗i , fi∗ . Assume (B, ei , fi ) and (B, e∗i , fi∗ )
are both lowest weight combinatorial crystals with the same lowest weight element b− , where
the other data is determined by setting wt(b− ) = 0. Assume further that, for all i 6= j ∈ I
and all b ∈ B,
(i)
(ii)
(iii)
(iv)
(v)
(vi)
ei (b), e∗i (b) 6= 0.
e∗i ej (b) = ej e∗i (b),
For all b ∈ B, ϕi (b) + ϕ∗i (b) − hwt(b), αi∨ i ≥ 0
If ϕi (b) + ϕ∗i (b) − hwt(b), αi∨ i = 0 then ei (b) = e∗i (b),
If ϕi (b) + ϕ∗i (b) − hwt(b), αi∨ i ≥ 1 then ϕ∗i (ei (b)) = ϕ∗i (b) and ϕi (e∗i (b)) = ϕi (b).
If ϕi (b) + ϕ∗i (b) − hwt(b), αi∨ i ≥ 2 then ei e∗i (b) = e∗i ei (b).
then (B, ei , fi ) ' (B, e∗i , fi∗ ) ' B(−∞).
2.3. Some topology. All our topological spaces are the F points of algebraic varieties for
some infinite field F, which is typically not algebraically closed. The topology is the Zariski
topology; that is, closed sets are locally defined as the zero sets of some polynomials.
Since we work over non-algebraically closed fields, All the varieties we study decompose
In fact, our varieties are all birationally equivalent to Fk for various k.
As usual, we say a space X is irreducible if it cannot be written as the union of two
proper closed subsets. In that case, we say the dimension of X is the maximal d such that
there is a sequence of irreducible subsets
(9)
∅ ⊂ X0 ⊂ · · · ⊂ Xd = X
with all containments proper. When X is birational to Fk , the dimension is k.
If X is reducible, its irreducible components are the irreducible subsets which are not
properly contained in larger irreducible subsets.
The following is certainly well known.
Lemma 2.4. If π : X → Y is a locally trivial fiber bundle with irreducible fiber F , then
there is a bijection between the irreducible components of X and Y . If Y (or equivalently
X) is irreducible, then dim X = dim Y + dim F .
Proof. First assume that Y is irreducible; we will show that X is irreducible. Suppose
that X = X1 ∪ X2 , where X1 and X2 are closed in X. For each y ∈ Y , π −1 (y) =
(π −1 (y)∩X1 )∪(π −1 (y)∩X2 ); since π −1 (y) ' F is irreducible, π −1 (y) ⊆ X1 or π −1 (y) ⊆ X2 .
Let Y1 = {y|π −1 (y) ⊆ X1 }, Y2 = {y|π −1 (y) ⊆ X2 }. We will show that Y1 and Y2 are closed
sets; then since Y = Y1 ∪ Y2 is irreducible it will follow that Y = Y1 or Y = Y2 ; and hence
X = X1 or X = X2 as required.
Trivialize π over an open cover {Y i } of Y . We will prove that Y1 ∩ Y i is closed in Y i for
each i; this implies that Y1 is closed (since then Y˜1 = ∪(Y˜1 ∩ Y i ) is open, where Y˜1 is the
CRYSTALS AND MODULATED GRAPHS
7
complement of Y1 ). Now identifying π −1 (Y i ) ' Y i × F , we have
Y1 ∩ Y i = {y ∈ Y i |y × F ⊆ X1 } = ∩f ∈F {y ∈ Y i |y × f ∈ X1 }.
(10)
This is closed, since {y ∈ Y i |y × f ∈ X1 } is closed (it is the pre-image of X under the map
Y i → Y i × F, y → (y, f )). Similarly Y2 is closed in Y .
Now if Y is not irreducible, suppose Y = ∪i Yi is it’s decomposition into irreducible
components; then using the above argument, π −1 (Yi ) is irreducible, so X = ∪i π −1 (Yi ) is
its decomposition into irreducible components.
3. Representation varieties and crystals
3.1. Definition of the representation variety. Fix a modulated graph M with Cartan
matrix C, as defined in §1.2. Denote the corresponding tensor algebra and preprojective
algebra
by T and Λ respectively (see Definition 1.1). Fix an I-graded vector space V =
P
V
,
where, as in §1.3, Vi is a vector space over Fi , and let T (v) and Λ(v) be the
i
i,∈I
corresponding varieties of nilpotent representation as in §1.3. We associate to v = dim V
a point in Q+ by
X
(11)
dim V =
vi αi ,
i∈I
and sometimes abuse notation by using, for example, (v, v) to mean (
P
i∈I
vi αi ,
P
i∈I
vi αi ).
Example 3.1. Continue considering the modulated quiver from Example 1.2. By definition, an element of T (α1 + α2 ) is a choice of
• 2 x1 ∈ HomC (C ⊗R R, C) = HomC (C, C) = C. Call this z = z1 + zi i.
• 1 x2 ∈ HomR (C ⊗C C, R) = HomR (C, R). Denote this by (w1 , wi ) ∈ R2 , where the
homomorphism sends 1 to w1 and i to wi .
So, T (α1 + α2 ) is isomorphic to R4 .
The relations cutting out Λ are
1 x2
(12)
2 x1
(13)
◦ 2 x1 = 0 ⇔ 1 x2 ◦ 2 x1 (1) = 0 ⇔ w1 z1 + wi zi = 0,
◦ 1 x2 − i2 x1 ◦ 1 x2 i = 0
⇔
w1
wi
and
z1 zi
−
w1
0 −1
0 −1
z1 zi
=0
1 0
wi
1 0
⇔ z1 w1 + z2 w2 = 0, z2 w1 − z1 w2 = 0.
The real solutions to these equations form two components, defined by {w1 = wi = 0} and
{z1 = zi = 0} (equivalently, by 1 x2 = 0 or 2 x1 = 0). However, as a real algebraic variety,
there would be a third component, defined by the equations {x2i = −x21 , y11 = −y22 , x1 y1 =
−x2 y2 }. This last component contains no new real points, and if we base change to C would
decompose further into two components: {x1 = x2 i, y1 = y2 i} and {x1 = −x2 i, y1 = −y2 i}.
This demonstrates both why we need to work with the space of F-points as opposed to
the abstract algebraic variety.
8
VINOTH NANDAKUMAR AND PETER TINGLEY
3.2. Some important spaces.
Definition 3.2. Fix i ∈ I. Set V i =
M
i Mj
⊗Fj Vj .
j:j ai ∈A
Definition 3.3. For j ai ∈ A, define
ιa : Vi → i Mj ⊗Fj j Mi ⊗Fi Vi
v → rij ⊗ v,
where rij is the canonical element as from (2). Define j x̃i : Vi → i Mj ⊗Fj Vj to be the
composition of ιa with the map
Definition 3.4. x̃i =
M
1 ⊗ j xi : i Mj ⊗ j Mi ⊗ Vi → i Mj ⊗ Vj .
M
i
i
and i x̃ =
j x̃i : Vi → V ,
i xj : V → Vi .
j∈I
j∈I
Proposition 3.5. The maps x̃i and i x̃ are both Fi linear.
Proof. The map ιa intertwines the left Fi module structure on Vi with the left Fi module
structure on Ma , which immediately implies that each j x̃i is Fi linear, so by summing x̃i
is as well. That i x̃ is Fi linear is immediate from the definition.
Definition 3.6. For each i, let Si be the simple Λ module such that ei Si = Si and
dimFi ei Si = 1 (so Si is a copy of Fi lying over vertex i, and all j xi are 0).
Lemma 3.7. Fix a representation V of Λ. Then Hom(Si , V ), Hom(V, Si ) and Ext1 (Si , V )
are all naturally Fi vector spaces, with
• Hom(Si , V ) ' ker x̃i
• Hom(V, Si ) ' (Vi / im i x̃)∗
• dimFi Ext1 (Si , V ) = dimFi V i − dimFi im x̃i − dimFi im i x̃.
Proof. The first two statements are obvious. For the third, we seek to classify extensions
ι
f
0 → Si →
− V0 −
→V →0
up to equivalence. Clearly Vj0 = Vj if j 6= i. Choose a vector space splitting Vi0 ' Vi ⊕ Fi ,
where Fi = im ι. The extension is uniquely determined by a map φ : V i → Fi subject to
the condition that the composition
(x̃i ,0)
(i x̃,φ)
Vi ⊕ Fi −−−→ V i −−−→ Vi ⊕ Fi
is 0. This precisely says that that ker φ ⊃ im x̃i , so φ ∈ Hom(V i /im x̃i , Fi ).
Two maps φ, φ0 give rise to the same class in Ext1 (Si , V ) if there exists a map
θ : Vi ⊕ Fi → Vi ⊕ Fi
which is the identity on Fi and on (Vi ⊕ Fi )/Fi , and such that
(i x̃, φ0 ) = θ ◦ (i x̃, φ)
Such maps are exactly (v, x) → (v, x + κ(v)) for linear κ : Vi → Fi , and such a map
stabilizes the short exact sequence iff ker κ ⊃ im i x̃. Thus the orbit of a short exact
sequence is parameterized by κ|im i x̃ , and the result follows.
CRYSTALS AND MODULATED GRAPHS
9
Lemma 3.8. dimFi Ext1 (Si , V ) = dimFi Hom(Si , V ) + dimFi Hom(V, Si ) − hdim V, αi i.
Remark 3.9. For Lusztig’s preprojective algebra, Lemma 3.8 still holds if Si is replaced
by an arbitrary finite dimensional module W (see [C-B00, Lemma 1]). However, for Dlab
and Ringel’s preprojective algebras, this more general statement is false (see §4.2).
Proof. Using Lemma 3.7, we compute as follows:
dim Ext1 (Si , V ) + hdimV, αi i − dim Hom(Si , V ) − dim Hom(V, Si )
= dim(V i /im(x̃i )) − dim(im(i x̃)) + hdimV, αi i − dim(ker(x̃i )) − dim(Vi /im(i x̃))
= dim(V i ) − dim(ker(x̃i )) − dim(im(x̃i )) + hdimV, αi i − dim(Vi )
= dim(V i ) − 2 dim(Vi ) + hdimV, α̌i i = 0
To see the last equality, note that hαj , α̌i i = ci,j = dimFi (i Mj ).
3.3. Relations between components. Define a function on Λ(v) by
(14)
ϕi (x) = dimFi ker xi .
Certainly ϕi is constructible. Let Λ(v)i;k be the subset of Λ(v) where ϕi takes the value k.
Fix v and let v̄ = v − kαi . Fix vector spaces V, V̄ of graded dimensions v, v̄, such that
V̄j = Vj for all j 6= i. Let Λ(v; i; k) be the variety whose points consist of an element of
Λ(v) along with a short exact sequence
0 → Fki → V → V̄ → 0,
(15)
which is trivial on Vj for all j 6= i. More explicitly, Λ(v; i; k) is the set of triples (x, P, Q)
where x ∈ Λ(V )i;k , P : Vi 7→ V̄i , Q : Fik 7→ V , and
• P is the identity on Vj for all j 6= i.
• ker P = ker xi
• im Q = ker xi .
Consider the obvious projections
(16)
Λ(v)i;0 o
π3
π2
Λ(v̄)i;0 × Surj(Vi , V̄i ) o
Λ(v; i; k)
π1
/ Λ(v)i;k
Proposition 3.10. π1 is a locally trivial fiber bundle whose fibers are isomorphic to
GL(V̄i ) × GL(Fki ).
π2 is a locally trivial fiber bundle whose fibers are isomorphic to
V
GL(Fki ) × Hom(Fdim
i
i −dim V
i
, Fki ).
In particular, the fibers of π1 have dimension vi2 − 2vi k + 2k 2 and the fibers of π3 ◦ π2 have
dimension vi2 − 2vi k + 2k 2 + k dimFi V i over Fi .
Proof. First consider π1 . Choose an Fi subspace W of Vi of co-dimension k and isomorphisms M : Fki → Vi /W and N : V̄i → W. Then on the locus where W ∩ ker xi = 0 (which
is a condition on the base) the map
(17)
(x, P, Q) → (x, (P ◦ N, M ◦ πker xi ◦ Q))
10
VINOTH NANDAKUMAR AND PETER TINGLEY
is the required local isomorphism, where πker xi is the projection that kills W . The local
inverse is
(x, (A, B)) → (x, A ◦ N −1 , πker xi ◦ M −1 ◦ B).
(18)
Now consider π2 . Fix W ∈ Vi of codimension k and L ⊂ V i of codimension r =
dim V i − dim V̄i , and choose Fi vector space isomorphisms M : Vi /W → Fki , ι : Fri → L.
Then, on the locus where M ∩ ker P = 0 and L ∩ ker i x̃ = 0 (a condition on the base),
(19)
(x, P, Q) → (x̄, P ) , M ◦ πker xi ◦ Q, Q−1 ◦ πker xi ◦ i x ◦ ι
is the necessary local isomorphism, where both πker xi and πker x0i are the projections onto
these spaces which kill the chosen subspace W . The inverse sends ((x̄, P ), (R, γ)) to the
short exact sequence
(20)
/ Fk
0
i
πker xi ◦M −1
/V
P
/V
/0
along with the extension of (V̄ , x̄) to V = W + ker P defined by the map γ.
X
X
Definition 3.11. D(v) =
dimF (HomFi (Vi , Vi )) =
di vi2 .
i∈I
i∈I
Lemma 3.12. Λ(v) has pure dimension D(v) − 21 (v, v) (over F). Furthermore, each
Λ(v)i;k `
is also pure of this dimension, and for each i there is a bijection between IrrΛ(v)
and Irr k Λ(v)i;k which takes X to X ∩ Λ(v)i;k for the unique k for which this is dense in
X.
Proof. Proceed by induction on dim(v), the case v = 0 being trivial. Fix v, and assume
the statement for all smaller v̄.
Fix k > 1. By Proposition 3.10, Λ(v; i; k) is a fiber bundle over each of Λ(v)i;k and
Λ(v − k1i )i;0 . By lemma 2.4 gives the desired bijection of components, and furthermore by
considering the dimensions of the fibers we see that
dimF Λ(v)i;k
= dimF Λ(v − kαi )i;0 + di k dimFi V i
1
= D(v − kαi ) − (v − kαi , v − kαi ) + di k(2νi − hv, α̌i i)
2
1
(v, α̌i )
)
= {D(v) − 2di νi k + di k 2 } − {(v, v) − 2k(v, αi ) + 2di k 2 } + di k(2νi −
2
di
1
= D(v) − (v, v)
2
Now, fix an irreducible component of Λ(v). Every point in Λ(v) is nilpotent, so is in
Λ(v)i;k for some i and some k > 1. In particular every irreducible component of Λ(v) has
an open dense subset contained in Λ(v)i;k for some i and some k ≥ 1. Then the result
follows as above.
Finally we must handle the case of Λ(v)i;0 . But this is open in Λ(v), so every irreducible
component is open and dense in some irreducible component of Λ(v). So the result follows
by the previous paragraph.
The required bijections of components are then clear.
CRYSTALS AND MODULATED GRAPHS
11
3.4. Crystal operators. Let
(21)
B=
a
IrrΛ(v).
v
`
Lemma 3.12 shows that, for each i, there
` is a bijection between IrrΛ(v) and k IrrΛ(v)i;k ,
where X is sent to the component of k IrrΛ(v)i;k that is dense in X. Denote by Xo the
corresponding component. Furthermore, we have bijections
a
a
IrrΛ(v)i;0 .
IrrΛ(v)i;k →
(22)
f˜i;k :
v
v
Define
f˜i :=
(23)
G
−1 ˜
f˜i;k−1
fi;k ,
ẽi :=
k
G
−1 ˜
f˜i;k+1
fi;k .
k
using the bijection of components, this gives operators on B.
We also need the ∗ operators, which are constructed in a completely analogues way.
Define
(24)
ϕ∗i (x) = dimFi Vi /Im(i x)
Λ(V )ki = {x ∈ Λ(v) : ϕ∗i (x) = k}.
and
Let Λ∗ (v; i; k) = {(x, P ∗ , Q∗ ) where x ∈ Λ(V )ki , P : V 7→ V, Q : V 7→ Fki }
such that :
• P is the identity on Vj for all j 6= i,
• im P = im i x,
• ker Q = im i x.
We have projections
(25)
π1∗ : Λ∗ (v; i; k) → Λ(v)ki
and
π2∗ : Λ(v; i; k) → Λ(v̄)0i .
As with Λ(v; k), we find that
`
• There is a bijection X → X o between IrrΛ(v) and k IrrΛ(v)k .
∗ on the level of irreducible components.
• π2∗ ◦ (π1∗ )−1 defines a bijections f˜i;k
Define ∗ crystal operators by
G
G
∗
∗
∗
∗
.
)−1 ◦ f˜i;k
(26)
f˜i∗ = (f˜i;k−1
)−1 ◦ f˜i;k
and ẽ∗i = (f˜i;k+1
k
k
3.5. Reworded operators. The following reworded characterization of the crystal operators will be useful to us:
Proposition 3.13. Fix X ∈ IrrΛ(v). Then there is an open-dense subset Xo of X such
that, for all x ∈ Xo and all sufficiently generic extensions
0 → Si → (V 0 , x0 ) → (V, x) → 0,
(V 0 , x0 ) is in a single irreducible component Y ∈ IrrΛ(V 0 ), and Y = ẽi X. Furthermore, the
subset of Y which can be realized in this way is open-dense.
Similarly, there is an open-dense subset X o of X such that, for all x ∈ X o and all
sufficiently generic extensions
0 → (V, x) → (V 0 , x0 ) → Si → 0,
12
VINOTH NANDAKUMAR AND PETER TINGLEY
(V 0 , x0 ) is in a single irreducible component Y ∈ IrrΛ(V 0 ), and Y = ẽ∗i X. Furthermore, the
subset of Y which can be realized in this way is open-dense.
bo = X ∩ Λ(v)i;k where k = ε(X), and let Xo be the subset of X
bo consisting
Proof. Let X
bo is
of points which are not in any other irreducible components of IrrΛ(v). Clearly X\X
closed and Xo is non-empty, so since X is irreducible Xo is open dense. Consider the maps
(27)
Λ(v; k)
Λ(v + αi ; k + 1)
π1
z
Λ(v)i;k
π30 π20
π3 π2
&
π10
v
Λ(v − kαi )i;0
)
Λ(v + αi )i;k+1
Recall that all these maps give bijections on the level of irreducible components. Let X̄ be
the component of Λ(v − kαi ) corresponding to X, and X 0 the component of Λ(v + αi ). Let
bo be the subset of X which is in Λ(v)i;k and not in any other irreducible components of
X
b 0 be the subset of X 0 which is in Λ(v + αi )i;k and not in any other
Λ(v). Similarly let X
o
irreducible components of Λ(v + αi ). Then
(28)
bo ∩ π1 (π3 π2 )−1 π 0 π 0 (π 0 )−1 X
b0
Xo = X
3 2 1
o
is the desired set.
The second statement follows by a completely symmetric argument.
Lemma 3.14. Fix X ∈ B and T ∈ X generic. Then ϕi (X) + ϕ∗i (X) − hwt(X), αi∨ i =
Ext1 (T, Si ) = Ext1 (Si , T )
Proof. This is immediate from Lemma 3.8.
Corollary 3.15. Fix X ∈ B and i, j ∈ I.
(i) If i = j and ϕi (b) + ϕ∗i (b) − hwt(b), αi∨ i = 0, then ei (X) = e∗i (X)
(ii) If i = j and ϕi (b) + ϕ∗i (b) − hwt(b), αi∨ i ≥ 1, then ϕi (e∗i (X)) = ϕi (X) and
ϕ∗i (ei (X)) = ϕ∗i (X).
(iii) If either i 6= j or ϕi (b) + ϕ∗i (b) − hwt(b), αi∨ i > 1, then e∗i ej (X) = ej e∗i (X).
Proof. Fix X and let T be the representation corresponding to a generic point in X, meaning one where all ϕi , ϕ∗i are minimal.
In case (i), by Lemma 3.14, Ext1 (T, Si ) = Ext1 (Si , T ) = 0, so the generic extensions in
Proposition 3.13 are in fact trivial extensions, and ei (X) = e∗i (X).
In case (ii), dim Ext1 (T, Si ) > 0, so, if T 0 is the generic extension
(29)
0 → Si → T 0 → T → 0,
then, using Lemma 3.7, we get that Hom(T 0 , Si ) ' dim Hom(T, Si ), so ϕi (e∗i (X)) = ϕi (X)
from Proposition 3.13. The other equality is true by a similar argument.
In case (iii), consider a generic T 00 in e∗i ej X. First we describe why it suffices to see that
the natural homomorphism from the i-socle of T 00 to the j-head is trivial. By Proposition
3.13, applying either fi∗ fj or fj fi∗ generically takes a subquotient that decreases the dimension of both the i-head and j socle by 1, and the fact that the homomorphism from
CRYSTALS AND MODULATED GRAPHS
13
i-head to j-socle is trivial implies these operations commute. So,
fi∗ fj e∗i ej X = fj fi∗ e∗i ej X = X.
(30)
But fi∗ fj ej e∗i X = X as well so, since crystal operators are partial permutations, e∗i ej X =
ej e∗i X.
Now let us explain why the natural map from the i-socle of T 00 to the j-head is trivial.
When i 6= j this is clear; in the other case, if the map is non-zero, Si will occur as a direct
summand in T 00 (simply pick a copy of Si in the i-socle, whose image in the i-head is nonzero). Now note that T does not contain Si as a direct summand. If it did, let T = T ⊕ Si ;
since T was chosen to be generic, it follows that Ext1 (T , Si ) = 0, and hence Ext1 (T, Si ) = 0,
which we know to be false by Lemma 3.14. Consequently, a generic extension T 0 ∈ ei X
also doesn’t contain Si as a direct summand. To get the desired contradiction, note that a
generic T 00 ∈ e∗i T 0 doesn’t have Si as a direct summand either, since:
dim Ext1 (Si , T 0 ) = dim Ext1 (Si , T ) − 1 > 0
This equality is true using Lemma 3.14, combined with the prior observation that
ϕ∗ (ei (X)) = ϕ∗ (X) when ϕi (b) + ϕ∗i (b) − hwt(b), αi∨ i ≥ 1.
3.6. Realization
of B(−∞).
`
Let B = v Irr Λ(v). For each X ∈ Λ(v), define:
• The weight wt(X) = v (which as Defined in §3.1 is in the root lattice)
• ϕi (X) = min ϕi (x), εi (X) = ϕi (X) − hwt(X), αi∨ i.
x∈X
• ϕ∗i (X) = min ϕ∗i (x),
x∈X
ε∗i (X) = ϕ∗i (X) − hwt(X), αi∨ i.
Theorem 3.16. B, along with data above and the operators ẽi , f˜i from §3.4 is a realization
of B(−∞).
Proof. It is clear that B along with ∗ crystal operators from §3.4 is a combinatorial bicrystal.
Furthermore, the condition that any x ∈ Λ(v) is nilpotent implies that, for any X ∈ B,
there is some i and some j such that f˜i X, f˜j∗ X 6= 0. This, along with the definitions of
εi , ε∗i above, implies that B is a highest weight combinatorial crystal with respect to either
set of operators. It remains to check the conditions of Proposition 2.3. (i) is obvious from
definitions, and (ii)-(vi) are contained in Corollary 3.15.
4. Examples
4.1. Continuing type C2 . Consider again the modulated graph from Example 1.2, where
the two field are R and C. In this case Λ is representation-finite, and each indecomposable
representation can be uniquely identified by giving its socle filtration, which we record from
right to left. So, for example, CR2 means the unique indecomposable with a copy of the
simple C over vertex 2 in its head, and two copies of the simple R over vertex 1 in its socle.
The following are all the isomorphism classes of indecomposable Λ modules, where we draw
lines between pairs that admit a non-trivial extension.
14
VINOTH NANDAKUMAR AND PETER TINGLEY
R
RC
C
RCR
CR2
CR
CR2 C
R2 C
To show that there are no other indecomposables, we argue as follows. One readily
verifies that RCR and CR2 C are indecomposable projectives. Suppose we have a representation M, consisting of a real vector space V , a complex vector space W , and maps
α : V → W, β : W → V satisfying the modulated pre-projective relations. If we can find
a path of length two which acts non-trivially on some element in V or W , it is easy to see
that the sub-module generated by that element is isomorphic to either RCR, or CR2 C, and
hence occurs as a direct summand. Else, if all paths of length two act as 0, then one can
break up the module as a direct sum M1 ⊕ M2 , where τ |M1 = 0, and τ |M2 = 0.
In each irreducible component of Λ(v), the isomorphism class of the corresponding representation is constant on an open-dense set, and the classes that show up this way are exactly
the rigid representations; that is representations such that no two indecomposables in their
Krull-Schmidt decomposition admit a non-trivial extension. So, the irreducible components of Λ(v) correspond to collections of indecomposables none of which are connected by
lines in the list above, whose total dimension in v. For example, consider v = (3, 2). The
number of Kostant partitions of 3α1 + 2α2 is 5, so the crystal-theoretic result above implies
there must be 5 irreducible components of Λ(v). The corresponding rigid modules are:
(R)(CR2 C), (CR)(CR2 ), (CR)(RCR), (RC)(R2 C), (RC)(RCR).
Unfortunately, this method does not generalize, since usually the preprojective algebra has
infinitely many isomorphism classes of indecomposables, even if the Cartan matrix is of
finite type.
b 2 . Dlab and Ringel’s construction can give
4.2. Deformed construction over C for sl
non-standard preprojective algebras even when all the Fi are chosen to be C. An example
b n , n ≥ 3 (where all fields are C, and the bimodules are
of this is given in [Rin98, §6] for sl
b 2.
all rank 1). Here we will give a similar example for sl
Consider Γ = (I, E) where I consists of two vertices labeled 1, 2 and E consists of a single
edge joining these two vertices. Choose F1 , F2 = C, and 1 M2 = 2 M1 = C2 , with the actions
of both F1 and F2 given by scalar multiplication on both bimodules. The corresponding
b 2 . Define
Cartan matrix, as defined in the introduction, is that of sl
21 :
1 M2
⊗C 2 M1 → C
(v1 , v2 ) ⊗ (w1 , w2 ) → v1 w1 + v2 w2
and
12 :
2 M1
⊗C 1 M2 → C
(w1 , w2 ) ⊗ (v1 , v2 ) → w1 v1 − w2 v2
Fix a graded vector space V = V1 ⊕ V2 and x ∈ Λ(V ). Define
CRYSTALS AND MODULATED GRAPHS
15
• m(0,1) = 2 x1 ((0, 1) ⊗ ·), m(1,0) = 2 x1 ((1, 0) ⊗ ·) in Hom(V1 , V2 ),
• m(0,1) = 1 x2 ((0, 1) ⊗ ·), m(1,0) = 1 x2 ((1, 0) ⊗ ·) in Hom(V2 , V1 ).
These four maps determine x. The preprojective relations are
(31)
m(1,0) m(1,0) − m(0,1) m(0,1) = 0
and m(1,0) m(1,0) + m(0,1) m(0,1) = 0.
Now consider the case where V1 and V2 are both one dimensional, with bases {v1 }, {v2 }
respectively. We define a representation y ∈ Λ(v) by specifying the 4 maps as above:
• m(1,0) = m(0,1) is the map which sends v1 to v2 ,
• m(1,0) = m(0,1) = 0.
Call this module I. Take a second copy I 0 of I, where the basis vectors are v10 , v20 . Any
extension of I by I 0 will be determined by a, b, c, d ∈ C defined by
(32) m(1,0) (v10 ) = v20 + av2 , m(0,1) (v10 ) = v20 + bv2 , m(1,0) (v20 ) = cv1 , and m(0,1) (v20 ) = dv1 ,
where a, b, c, d ∈ C. The two preprojective relations give the equations:
(33)
c−d=0
and c + d = 0.
The only solution is c = d = 0, so any such extension has a two-dimensional head.
b 2 , if V is
A simple calculation shows that, for Lusztig’s preprojective algebra of type sl
any indecomposable and fits in a short exact sequence 0 → S2 → V → S1 → 0, then V has
a self-extension with a 1-dimensional head. Hence, for the choices made in this example,
Dlab and Ringel’s preprojective algebra is not isomorphic to Lusztig’s.
In fact, we can consider a one parameter family of preprojective algebras which are
defined as above but with
(34)
12 = zw1 v1 − w2 v2 .
For z = −1 we get exactly Lusztig’s preprojective algebra, but in all other cases the above
argument shows that we do not. So in this case Dlab and Ringel’s construction can be
though of as non-trivially deforming Lusztig’s preprojective algebras.
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[C-B00]
[DR80]
[Gab73]
[GLSa]
[GLSb]
[Kas93]
[Kas95]
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[Lus91]
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(2000), no. 5, 1027–1037.
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9-36. arXiv:q-alg/9606009
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4 (2):365–421, 1991.
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[Rin98]
[Sav05]
[TW]
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467480, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998.
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P. Tingley and B. Webster. Mirković-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras. Preprint. arXiv:1210.6921
Dept. of Mathematics, University of Utah
E-mail address: vinoth.90@gmail.com
Dept. of Math and Stats, Loyola University Chicago
E-mail address: ptingley@luc.edu