LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
4.
Deformed preprojective algebras, cont’d
4.1.
Recap.
Recall that in the previous lecture we have identified
C
⟨
x, y
⟩
#Γ with
T
C
Γ
(
C
2
⊗
C
Γ), and
C
Q
with
T
(
C
Q
)
0
(
C
Q
)
1
. Also recall that
f
C
⟨
x, y
⟩
#Γ
f
∼
=
C
Q
, where
f
=
⊕
r
i
=0
f
i
∈
C
Γ =
⊕
r
i
=0
End(
N
∗
i
) with
f
i
being a primitive idempotent in End(
N
i
)
∗
. Under this identi-
fication,
f
i
∈
C
Q
becomes the path
ϵ
i
.
Further, to
i
∈
Q
0
we have assigned an element [
a
∗
, a
]
i
∈
ϵ
i
(
C
Q
)
2
ϵ
i
by the formula
[
a
∗
, a
]
i
=
⊕
a
∈
Q
1
,t
(
a
)=
i
a
∗
a
−
∑
a
∈
Q
1
,h
(
a
)=
i
aa
∗
.
Also to
c
∈
(
C
Γ)
Γ
we assign
λ
= (
λ
i
)
i
∈
Q
0
by
λ
i
= tr
N
i
c
. The main result we are going to
prove is a theorem of Crawley-Boevey and Holland.
Theorem 4.1.
The ideal
f
(
xy
−
yx
−
c
)
C
⟨
x,y
⟩
#Γ
f
is generated by the elements
[
a
∗
, a
]
i
−
λ
i
ϵ
i
, i
∈
Q
0
.
A key step in the proof is the following lemma again due to Crawley-Boevey and Holland.
Lemma 4.2.
To each
a
∈
Q
1
one can associate
η
a
∈
Hom
Γ
(
N
t
(
a
)
,
C
2
⊗
N
h
(
a
)
)
, θ
a
∈
Hom
Γ
(
N
h
(
a
)
,
C
2
⊗
N
t
(
a
)
)
that combine to form bases in the spaces
Hom
Γ
(
N
i
,
C
2
⊗
N
j
)
are all
i, j
and satisfy
(1)
∑
a
∈
Q
1
,t
(
a
)=
i
(1
C
2
⊗
θ
a
)
η
a
−
∑
a
∈
Q
1
,h
(
a
)=
i
(1
C
2
⊗
η
a
)
θ
a
=
δ
i
(
ζ
⊗
1
N
i
)
,
(the equality of maps
N
i
→
C
2
⊗
C
2
⊗
N
i
) for all
i
.
To prove the lemma we have introduced explicit mutually inverse isomorphisms of the
spaces Hom
Γ
(
M,
C
2
⊗
M
′
)
,
Hom
Γ
(
C
2
⊗
M, M
′
). Namely, we map
ψ
∈
Hom
Γ
(
M,
C
2
⊗
M
′
) to
ψ
♡
:= (
ω
⊗
1
M
′
)
◦
(1
C
2
⊗
ψ
)
∈
Hom
Γ
(
C
2
⊗
M, M
′
), and we map
φ
∈
Hom
Γ
(
C
2
⊗
M, M
′
) to
(1
C
2
⊗
φ
)
◦
(
ζ
⊗
1
M
)
∈
Hom(
M,
C
2
⊗
M
′
). Here
ω
is the skew-symmetric form on
C
2
given by
ω
(
y, x
) = 1 =
−
ω
(
x, y
) (and viewed as a map
C
2
⊗
C
2
→
C
) and
ζ
=
x
⊗
y
−
y
⊗
x
(viewed
as a map
C
→
C
2
⊗
C
2
).
4.2.
Proof of the CBH lemma.
From now on we concentrate on the non-cyclic case. A
special feature of this case is that Γ is a tree.
The spaces Hom
Γ
(
N
i
,
C
2
⊗
N
j
) when
i
=
t
(
a
)
, j
=
h
(
a
) or vice versa are 1-dimensional. For
a moment, choose arbitrary nonzero
η
a
, θ
a
, they are defined up to a nonzero scalar multiple.
Then
θ
♡
a
η
a
is a nonzero endomorphism of
N
t
(
a
)
, while
η
♡
a
θ
a
is a nonzero endomorphism of
N
h
(
a
)
. Multiplying
θ
a
by a nonzero scalar
k
, we also multiply those two endomorphisms by
k
. We claim that there are nonzero scalars
d
i
, i
∈
Q
0
,
with the property that (after rescaling
the
θ
i
’s) we get
(2)
θ
♡
a
η
a
=
d
h
(
a
)
1
N
t
(
a
)
, η
♡
a
θ
a
=
−
d
t
(
a
)
1
N
h
(
a
)
.
1