Iodine-Promoted Domino Oxidative Cyclization for the One-Pot Synthesis of Novel Fused Four-Ring Quinoxaline Fluorophores by sp3 C?H Functionalization

Article abstract of DOI:10.1002/chem.201903688

A method for the synthesis of novel fused four-ring quinoxaline skeleton has been described by an I₂ promoted sp³ C?H functionalization between 1,2,3,3-tetramethyl-3H-indolium iodides and 1,2-diamines. This transformation proceeds smoothly under metal- and peroxide-free conditions through a sequential iodination, oxidation, annulation and rearrangement. Moreover, 8,9-dichloro-5,12,12-trimethyl-2-(trifluoromethyl)-5,12-dihydroquinolino[2,3-b]quinoxaline showed good photophysical properties and was used in live cell imaging, indicating the potential value of this skeleton as a fluorophore in probes.

Full text of DOI:10.1002/chem.201903688

Journal of Algebra 527 (2019) 196–203
Contents lists available at ScienceDirect
Journal of Algebra
www.elsevier.com/locate/jalgebra
Toën’s formula and Green’s formula
Haicheng Zhang
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal
University, Nanjing 210023, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history:
Let A be a finitary hereditary abelian category. In this note,
we show that Green’s formula on Hall numbers of A can be
deduced from the associativity of the derived Hall algebra of
A whose multiplication structure constants are given by the
so-called Toën’s formula.
Received 24 August 2018
Available online 12 March 2019
Communicated by Volodymyr
Mazorchuk
©
2019 Elsevier Inc. All rights reserved.
MSC:
1
1
1
6G20
7B20
7B37
Keywords:
Derived Hall algebra
Toën’s formula
Green’s formula
1
. Introduction
The Hall algebra of a finite dimensional algebra over a finite field was introduced by
Ringel [6] in 1990. Ringel [7] proved that if A is a representation finite hereditary algebra,
then the twisted Hall algebra Hv(A), called the Ringel–Hall algebra, is isomorphic to
the positive part of the corresponding quantized enveloping algebra. After that, Green
[
3] generalised Ringel’s result to any hereditary algebra A by introducing a bialgebra
structure on Hv(A). It is worth mentioning that the key ingredient of the bialgebra
E-mail address: zhanghc@njnu.edu.cn.
https://doi.org/10.1016/j.jalgebra.2019.02.033
0
021-8693/© 2019 Elsevier Inc. All rights reserved.

H. Zhang / Journal of Algebra 527 (2019) 196–203
197
structure of Hv(A) is the so-called Green’s formula, whose original proof is given in
some complicated homological methods. Xiao [12] provided the antipode for Hv(A), and
obtained a realization of the whole quantized enveloping algebra by constructing the
Drinfeld double of the extended Ringel–Hall algebra of A.
Trying to give an intrinsic realization of the entire quantized enveloping algebra via
Hall algebra approach, Toën [11] defined a derived Hall algebra for a differential graded
category satisfying some finiteness conditions, whose multiplication structure constants
are given by the so-called Toën’s formula. Later on, Xiao and Xu [13] generalised Toën’s
result to any triangulated category satisfying certain finiteness conditions, which includes
the bounded derived category of a finitary hereditary abelian category.
As a matter of fact, Bridgeland [1] has recently given a beautiful realization of the en-
tire quantized enveloping algebra via the Hall algebra of 2-cyclic complexes of projective
modules over a hereditary algebra A, which is called the Bridgeland Hall algebra of A.
Yanagida [14] proved that the Bridgeland Hall algebra of a hereditary algebra is isomor-
phic to its Drinfeld double Hall algebra. Afterwards, Zhang [15] used the associativity
of the Hall algebra of A to give a simple proof of this result.
In order to generalise Bridgeland’s construction to any hereditary abelian category
which may not have enough projectives, Lu and Peng [5] introduced the modified Ringel–
Hall algebra, whose construction is much technical. Recently, Lin and Peng [4] used the
associativity of the modified Ringel–Hall algebra of bounded complexes of a hereditary
abelian category to give a proof of Green’s formula. However, for this they first give some
characterizations on the structure of the modified Ringel–Hall algebra, in particular, they
need to present a basis and some relations.
Actually, for a hereditary abelian category with enough projectives, it is similar to [4]
that we can also give a proof of Green’s formula via the associativity of the Bridgeland
Hall algebra of bounded complexes or m-cyclic (m > 2) complexes over projective objects,
which is defined in [2] and further studied in [16].
In this note, inspired by the work in [4], we want to disclose the relation between
Green’s formula and Toën’s formula, which are two significant formulas in the Hall alge-
bra and derived Hall algebra, respectively. Explicitly, we will prove that Toën’s formula
“
implies” Green’s formula. That is, we will use the associativity of the derived Hall al-
gebra to give a proof of Green’s formula, without involving in the modified Ringel–Hall
algebras or Bridgeland Hall algebras. Compared with the modified Ringel–Hall algebra,
the basis of the derived Hall algebra is immediate, and the relations are also well-known.
So it is more direct to use the derived Hall algebra to give a proof of Green’s formula.
Throughout the paper, k is a finite field with q elements, A is a hereditary abelian
b
k-category, and D (A) is the bounded derived category of A . Let K(A) be the
ˆ
Grothendieck group of A , and for any M ∈ A we denote by M the image of M in
K(A). For a finite set S, we denote by |S| its cardinality. For any object X in A or
b
D (A ), we denote by [X] the isoclass of X, by Aut X the automorphism group of X,
and write aX for |Aut X|.

1
98
H. Zhang / Journal of Algebra 527 (2019) 196–203
2
. Preliminaries
A small abelian k-category A is called finitary if HomA (M, N) and Ext 1 (M, N) are
A
both finite dimensional for any objects M, N ∈ A . In what follows, A is always assumed
to be finitary.
2.1. Hall algebras
Given X, Y, L ∈ A , let
ϕ
ψ
L
XY
W
:= {(ϕ, ψ) | 0 → Y −→ L −→ X → 0 is exact in A }.
L
XY
The group G := Aut X × Aut Y acts on W
via
ϕ
ψ
ψ
0
0
Y
Y
L
X
X
0
g
f
ϕ
L
0.
L
XY
−1
That is, for any (ϕ, ψ) ∈ W
XY
and (f, g) ∈ G, (f, g) · (ϕ, ψ) = (ϕg , fψ). We denote
L
the set of G-orbits by V . Since ϕ is monic and ψ epic this action is free, and we define
L
XY
|
W
|
.
L
XY
L
XY
g
:= |V | =
aXaY
Definition 2.1. The Hall algebra H (A ) of A is the vector space over C with basis
| X ∈ A } indexed by the set of isoclasses of objects in A , and with the mul-
{
u
[X]
tiplication defined by
ꢀ
L
u
u
=
g
u
.
[
X] [Y ]
XY [L]
[
L]
It is well-known that the Hall algebra H (A ) is an associative and unital algebra.
2.2. Derived Hall algebras
b
Given X, Y, L ∈ D (A ), let
f
g
h
L
XY
b
W
:= {(f, g, h) | Y −→ L −→ X −→ Y [1] is a triangle in D (A )}.
L
XY
Consider the action of Aut Y on W
defined by

H. Zhang / Journal of Algebra 527 (2019) 196–203
199
f
g
h
Y
Y
L
L
X
X
Y [1]
η
η[1]
ꢀ
ꢀ
h
f
g
Y [1]
L
XY
∗
Y
with η ∈ Aut Y . Denote the set of orbits by (W ) . Dually, we consider the action of
L
XY
L
XY
∗
Aut X on W
and obtain the orbit set (W ) .
X
Since the actions above are not free, in general,
L
XY
∗
L
XY X
∗
|
(W ) |
|(W ) |
.
Y
ꢀ
aX
aY
However, by [11,13] we know that
L
XY
∗
L
XY
∗
|
(W ) | {L, X}
|(W ) | {Y, L}
Y
X
L
·
=
·
=: F
,
XY
aX
{X, X}
aY
{Y, Y }
for any M, N ∈ D (A ). In fact, it is
ꢁ
i
−1)
b
where {M, N} :=
easy to see that
|Hom
b
(M[i], N)|(
D (A )
i>0
L
XY
∗
L
XY X
∗
|
(W ) | = |Hom
b
(L, X)Y [1]
|
and |(W ) | = |Hom
b
(Y, L)X|,
Y
D (A )
D (A )
where Hom
b
(M, N)Z denotes the subset of Hom
b
(M, N) consisting of mor-
D (A )
D (A )
phisms M → N whose cone is isomorphic to Z. Namely, we have the following formula
|Hom
b
(L, X)Y [1]
aX
|
{L, X}
{X, X}
|Hom
b
(Y, L)X| {Y, L} ,
D (A )
D (A )
L
XY
F
=
·
=
·
aY
{Y, Y }
which is called Toën’s formula. It is easy to see that for any X, Y, L ∈ A we have that
L
XY
L
XY
F
= g
.
Definition 2.2. The derived Hall algebra DH(A ) of A is the vector space over C with
b
b
basis {u
| X ∈ D (A )} indexed by the set of isoclasses of objects in D (A ), and with
[
X]
the multiplication defined by
ꢀ
L
u
u
=
F
u
.
[
X] [Y ]
XY [L]
[
L]
By [11,13], we know that DH(A ) is an associative and unital algebra.
For the convenience of the calculation in the sequel, let us introduce the Euler form.
Given objects M, N ∈ A, define
1
ꢁ
M, Nꢂ := dim kHomA(M, N) − dim kExt (M, N).
A

2
00
H. Zhang / Journal of Algebra 527 (2019) 196–203
It descends to give a bilinear form
·, ·ꢂ : K(A) × K(A) −→ Z,
ꢁ
known as the Euler form.
. Main results
The following formulas can be viewed as generalised versions of Green’s formula. It
3
is worth reminding that all the following sums are taken over the isoclasses of objects
in A .
ꢀ
ꢀ
Theorem 3.1. Given M, N, M , N , Y ∈ A , we have the following two formulas
1)
(
ꢀ
ꢀ
aP
L
N
L
aM aN aMꢀ
g
g
g
ꢀ
MN P Y M P
aL
[
L],[P ]
ꢀ
1
ꢀ
|
HomA (A, Y )| |Ext (A, B )|
ꢀ
ꢀ
ꢀ
N
A
M
B
N
M
=
g
ꢀ
g
ꢀ
g
ꢀ
g
g
ꢀ
ꢀ
aAaAꢀ aBaP ꢀ
;
AA P Y BP AB A B
ꢀ
1
|
HomA (A, B )| |Ext (A, Y )|
ꢀ
ꢀ
ꢀ
A
[
A],[A ],[B],[B ],[P ]
(
3.1)
(
2)
ꢀ
ꢀ
aP
M
L
L
aM aN aNꢀ
g
g
ꢀ
g
MN P N Y P
aL
[
L],[P ]
ꢀ
ꢀ
1
ꢀ
|
HomA (Y, B )| |Ext (A, B )|
ꢀ
ꢀ
N
A
M
A
N
M
=
g
ꢀ
ꢀ
g
ꢀ
g
ꢀ
g
g
ꢀ
ꢀ
aP ꢀ aAꢀ aBaBꢀ
.
ꢀ
1
ꢀ
P A Y P BB AB A B
|
HomA (A, B )| |Ext (Y, B )|
ꢀ
ꢀ
ꢀ
A
[
A],[A ],[B],[B ],[P ]
(
3.2)
ꢀ
ꢀ
Corollary 3.2. (Green’s formula) Given M, N, M , N ∈ A , we have the following for-
mula
ꢀ
1
L
L
aM aN aMꢀ aNꢀ
g
g
ꢀ
ꢀ
MN M N
aL
[
L]
(
3.3)
ꢀ
1
ꢀ
|
Ext (A, B )|
ꢀ
ꢀ
N
A
M
N
M
=
g
ꢀ
g
ꢀ
g
g
ꢀ
ꢀ
aAaAꢀ aBaBꢀ
.
AA BB AB A B
ꢀ
|
HomA (A, B )|
ꢀ
ꢀ
[
A],[A ],[B],[B ]
Proof. Taking Y = 0 in Theorem 3.1 gives the desired formula.
2
Before proving Theorem 3.1, we give two necessary lemmas.
The first lemma is well-known (for example, see [8, Lemma 3.15] for a proof).

H. Zhang / Journal of Algebra 527 (2019) 196–203
201
Lemma 3.3. Let f : X → Y be a morphism in A . Then f fits into the following triangle
b
in D (A )
f
X
Y
Ker (f)[1] ⊕ Coker (f)
X[1].
Given M, N, X, Y ∈ A , let
ϕ
f
ψ
XY
NM
W
:= {(ϕ, f, ψ) | 0 → X −→ M −→ N −→ Y → 0 is exact in A },
and set
XY
NM
|
W
|
.
XY
NM
γ
:=
(3.4)
(3.5)
aM aN
The following lemma is crucial to the proof of Main theorem.
Lemma 3.4. Given M, N, X, Y ∈ A , we have that
ꢀ
aXaY
X[1]⊕Y
M[1]N
X⊕Y [−1]
MN[−1]
−ꢁY,Xꢂ
M
LX Y L
N
F
= F
= q
g
g
aL.
aM aN
[
L]
Proof. By [11, Proposition 7.1], we know that
X[1]⊕Y
M[1]N
XY
NM
· q−ꢁY,Xꢂ
F
= γ
.
(3.6)
(3.7)
By [10, (8.8)],
ꢀ
aXaY
XY
NM
M
N
γ
=
g
g aL.
LX Y L
2
aM aN
[
L]
b
Proof of Theorem 3.1. (1) By abuse of notation, in what follows, for each X ∈ D (A ),
we write uX for u
.
X]
[
ꢀ
∼
ꢀ
On the one hand, by Hom
b
(N [1], L[1]) = Hom
b
(N , L) and Lemma 3.3, for
D (A )
D (A )
ꢀ
b
each f ∈ Hom
b
(N , L), it fits into the following triangle in D (A )
D (A )
f
ꢀ
Y [1] ⊕ Mꢀ
ꢀ
N
L
N [1],
ꢀ
where Y = Ker (f) and M = Coker (f). That is, every middle term of the extension of
ꢀ
ꢀ
L by N [1] is of the form Y [1] ⊕ M . Thus,
ꢀ
ꢀ
ꢀ
L
L
Y [1]⊕M
u
ꢀ
(uM uN ) =
g
u
ꢀ
uL =
g F
MN
ꢀ
u
Y [1]⊕Mꢀ
.
(3.8)
N [1]
MN N [1]
N [1]L
ꢀ
[
L]
[L],[Y ],[M ]

2
02
H. Zhang / Journal of Algebra 527 (2019) 196–203
Since A is hereditary, we obtain that
ꢀ
Y [1]⊕M
uMꢀ
u
= F
u
Y [1]⊕Mꢀ
.
(3.9)
Y [1]
ꢀ
M Y [1]
ꢀ
Y [1]⊕M
ꢀ
ꢀ
By [9, Lemma 2.2], we have that F
ꢀ
= {Y [1], M } · |Hom
b
(Y [1], M )| = 1. So
M Y [1]
D (A )
uMꢀ
u
= u
Y [1]⊕Mꢀ
, and thus
Y [1]
ꢀ
ꢀ
L
MN
Y [1]⊕M
u
ꢀ
(uM uN ) =
g
F
ꢀ
uMꢀ
u .
Y [1]
(3.10)
N [1]
N [1]L
ꢀ
[
L],[Y ],[M ]
On the other hand,
ꢀ
ꢀ
B [1]⊕A
(
u
ꢀ
uM )uN =
F
ꢀ
uAu
ꢀ
uN
N [1]
B [1]
N [1]M
ꢀ
[
[
[
B ],[A]
ꢀ
ꢀ
ꢀ
B [1]⊕A Y [1]⊕B
=
=
F
F
ꢀ
uAuBu
Y [1]
N [1]M
B [1]N
(3.11)
ꢀ
B ],[A],[Y ],[B]
ꢀ
ꢀ
ꢀ
ꢀ
B [1]⊕A Y [1]⊕B
M
F
F
ꢀ
g
uMꢀ
u
Y [1].
N [1]M
B [1]N AB
ꢀ
ꢀ
B ],[A],[Y ],[B],[M ]
ꢀ
Since u
ꢀ
(uM uN ) = (u
ꢀ
uM )uN , for each fixed M , Y , we have that
N [1]
N [1]
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
M
L
Y [1]⊕M
B [1]⊕A Y [1]⊕B
g
F
ꢀ
=
F
ꢀ
F
ꢀ
g
.
(3.12)
MN N [1]L
N [1]M
B [1]N AB
ꢀ
[
L]
[B ],[A],[B]
By Lemma 3.4, we obtain that
ꢀ
ꢀ
a a
Y
ꢀ
ꢀ
N
−
ꢁM ,Y ꢂ
L
MN
M
L
q
g
g
g
ꢀ
aP =
P Y M P
aNꢀ aL
[
L],[P ]
(
3.13)
ꢀ
ꢀ
aB
ꢀ
aA
ꢀ
ꢀ
a a
Y
ꢀ
ꢀ
M
−
ꢁA,B ꢂ−ꢁB,Y ꢂ
N
M
B
B
N
q
g
ꢀ
g
ꢀ
aAꢀ
g
ꢀ
g
ꢀ
aP ꢀ
g
.
AB
A B AA
P Y BP
aNꢀ aM
aBꢀ aN
ꢀ
ꢀ
ꢀ
[
B ],[A],[B],[A ],[P ]
Noting that
ꢀ
ꢀ
ˆ ꢀ ˆ ˆ
ꢀ
ꢁ
M ,Y ꢂ−ꢁA,B ꢂ−ꢁB,Y ꢂ
ꢁM −B,Y ꢂ−ꢁA,B ꢂ
ˆ ˆ ˆꢀ
q
= q
ꢁA,Y −B ꢂ
q
=
=
1
ꢀ
|
HomA (A, Y )| |Ext (A, B )|
A
,
ꢀ
1
|
HomA (A, B )| |Ext (A, Y )|
A
we complete the proof.
(
2) Using (uM uN )u
ꢀ
= uM (uN u
ꢀ
), we can prove it in a similar way as
M [−1]
M [−1]
(
1).
2

H. Zhang / Journal of Algebra 527 (2019) 196–203
203
Acknowledgments
This work is inspired by Ji Lin and Liangang Peng’s work in [4]. The author is pleas-
antly informed that Jie Xiao and Fan Xu have also known this idea after the paper of Lin
and Peng, but they do not write out. The author is grateful to the anonymous referee
for his/her valuable suggestions and comments, and supported by the National Natural
Science Foundation of China (No. 11801273), Natural Science Foundation of Jiangsu
Province of China (No. BK20180722) and Natural Science Foundation of Jiangsu Higher
Education Institutions of China (No. 18KJB110017).
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