Publicacions Ma em`a iques, Vol 39 (1995), 349–354.
ON THE DEFINITION
OF THE DUAL LIE COALGEBRA
OF A LIE ALGEBRA
Be in Dia a
Abs ac
Le Lbe a Lie algeb a o e a field K. The dual Lie coalgeb a L◦
o Lhas been defined by W. Michaelis o be he sum o all good
subspaces Vo he dual space L∗o L:Vis good i m(V)⊂
V⊗V, whe e mis he mul iplica ion o L. We show ha L◦=
m−1(L∗⊗L∗) as in he associa i e case.
Le Lbe a Lie algeb a o e he field Kwi h mul iplica ion m:L⊗L→
L: i.e. mis a linea map and se ing m(x⊗y)=[x, y], one has
(1) [x, x]=0
(2) [x, [y,z]]+[z,[x, y]] + [y, [z,x]] = 0.
Le L∗be he dual ec o space o Land m:L∗→(L⊗L)∗be he
anspose o m. We iden i y L∗⊗L∗wi h a subspace o (L⊗L)∗and
we se L= m−1(L∗⊗L∗).
Fix ∈L∗and conside he linea map γ :L→L∗defined by
(3) γ (x),y= ,[x, y]= m( ),x⊗y,x,y∈L.
Se ing, as usual, adx(y)=[x, y], one has γ (x)= (adx)( ). Some imes,
we shall w i e γ (x)=x· .
350 B. Dia a
Lemma 1. Fo ∈L∗, he ollowing s a emen s a e equi alen
(i) ∈L
(ii) The linea map γ :L→L∗is o fini e ank.
P oo : The equi alence ollows eadily om (3).
Mo eo e , since L∗⊗L∗can be iden ified wi h he space o he linea
maps o Lin o L∗o fini e ank, ia ι:L∗⊗L∗→Hom(L, L∗)by
se ing ι( ⊗g)(x)= (x)g, one has ∈Liff γ =
n

j=1
j⊗gjiff
m( )=
n

j=1
j⊗gj.
Lemma 2. Fo ∈Land x∈L, one has, γ (x)=x· ∈L.
Mo eo e γ (L)is a ec o subspace o Lo fini e dimension.
P oo : I ∈L, hen o x,y∈L, one has  ,[x, y]=γ (x),y=
n

j=1
 j,xgj,y. Howe e , (2) can be w i en [x, [y,z]] = −[y, [z,x]] −
[z,[x, y]]. The e o e, one has
γx· (y),z=x· ,[y,z]
= ,[x, [y,z]]
=− ,[y,[z,x]]− ,[z,[x, y]]
=−
n

j=1
 j,ygj,[z,x]−
n

j=1
 j,zgj,[x, y]
=
n

j=1
 j,yx·gj,z−
n

j=1
x·gj,y j,z.
Hence γx· (y)=
n

j=1
 j,yx·gj−
n

j=1
x·gj,y j. I ollows ha γx·
is o fini e ank, ha is x· =γ (x)∈L. Then, i is clea ha γ (L)
is a ec o subspace o Land fini e dimensional.
Dual Lie coalgeb a o a Lie algeb a 351
No e. One deduces om he abo e p oo ha i ∈Land m( )=
n

j=1
j⊗gj hen o x∈L, one has
m(x· )=
n

j=1
j⊗(x·gj)−
n

j=1
(x·gj)⊗ j.
Theo em 1. Lis a good subspace o L∗i.e. m(L)⊂L⊗L.
Mo eo e , one has L=L◦.
P oo : Le ∈Land le (gj)1≤j≤nbe a base o γ (L)⊂L. One
has gj=xj· and o any x∈L,γ (x)=
n

j=1
j(x)gj; hence m( )=
n

j=1
j⊗gj, j∈L∗. Howe e m=−m◦τ(skew-symme y), he e o e,
one has m=− τ◦ mand m( )=
n

j=1
j⊗gj=− τ(
n

j=1
j⊗gj)=
−
n

j=1
gj⊗ j. Since (gj)1≤j≤nis ee in L∗, he e exis s o 1 ≤≤n
y∈Lsuch ha gj,y
=δj. Hence (y⊗1L∗)( m( )) =
n

j=1
 j,y
gj=
−
n

j=1
gj,y
 j=− , ha is =−
n

j=1
 j,y
gj∈γ (L). I ollows
ha m( )=
n

j=1
j⊗gj∈γ (L)⊗γ (L)⊂L⊗Land L⊂L◦.
On he o he hand, i is clea ha any good subspace Vo L∗is
con ained in L, he e o e L◦⊂L. We ha e p o ed ha L=L◦.
No e. I ∈Land i (xj· )1≤j≤nis a base o γ (L), one has
m( )=
n

j=1
(yj· )⊗(xj· ) whe e, o 1 ≤j≤n,yjis such ha
 ,[x,y
j]=δj.
Fu he mo e, i x∈L, one has
(4) m(x· )=
n

j=1
(yj· )⊗[x·(xj· )] −
n

j=1
[x·(xj· )] ⊗(yj· ).
352 B. Dia a
Rema k. Pu ∆ = m|L:L→L⊗L. Following W. Michaelis
[1], (see also [2], [3], [4], [5] and [6]) one ob ains a Lie coalgeb a (L,∆),
ha is :
(5) ∆ = −τ◦∆
i he cha ac e is ic o Kis diffe en om 2 and Im ∆ ⊂Im(1L−τ)
o he wise [τ( ⊗g)=g⊗ ].
(6) (id3+σ+σ2)◦(1L⊗∆) ◦∆=0
whe e σ( ⊗g⊗h)=h⊗ ⊗g.
This ollows om (1) and (2). No ice ha (2) is equi alen o m◦
(1L⊗m)◦(id3+ρ+ρ2) = 0 whe e ρ(x⊗y⊗z)=z⊗x⊗yand one has
ρ2|L=σ.
Fo A⊂L, le span(A) be he ec o subspace o Lspaned by A.
Theo em 2. Le ∈L. Pu
V0=K·
V1=γ (L)={x1· ,x1∈L}
V2= span{x2· 1,x
2∈L, 1∈V1}
...................................................
Vn= span{xn· n−1,x
n∈L, n−1∈Vn−1}
...............................................................
Then W=
n≥0
Vnis a Lie subcoalgeb a o Land is he smalles Lie
subcoalgeb a o L ha con ains .
P oo : We ha e seen ha i ∈L, hen ∆( )=
n

j=1
(yj· )⊗(xj· ).
I ollows ha ∆(V0)⊂V1⊗V1. Fu he mo e V1⊂L, and by induc ion
one has Vn⊂L. On he o he hand, i xn∈L, n−1∈Vn−1,n≥1,
one has by (4)
∆(xn· n−1)=
m

j=1
(ynj · n−1)⊗[xn·(xnj · n−1)]
−
m

j=1
[xn·(xnj · n−1)] ⊗(ynj · n−1)∈Vn⊗Vn+1 +Vn+1 ⊗Vn.
The e o e, ∆(Vn)⊂Vn⊗Vn+1 +Vn+1 ⊗Vn⊂W⊗W,n≥1, and since
∆(V0)⊂V1⊗V1⊂W⊗W, one has ∆(W)⊂W⊗W, i.e. Wis a Lie
subcoalgeb a o L.
Dual Lie coalgeb a o a Lie algeb a 353
Le Vbe a Lie subcoalgeb a o L. Fo any h∈Vand x∈L, one
has ∆(h)=
n

j=1
h1
j⊗h2
j∈V⊗Vand x·h=γh(x)=
n

j=1
h1
j,xh2
j∈V.
The e o e, i Vcon ains , one has V0⊂Vand V1=γ (L)⊂V.I
is eadily seen by induc ion ha Vn⊂V o all n≥0. I ollows ha
W=
n≥0
Vnis con ained in V.
No e.
(i) One can p o e, by induc ion, ha he abo e Vn,n≥0, a e fini e
dimensional.
(ii) One has o n≥1, Vn= span{ ad x1◦ ad x2◦...◦ ad xn( ),
x1,... ,x
n∈L}.
The e o e, i Lis nilpo en o class k, hen o any ∈L, he associa ed
sequence o subspaces (Vn)n≥0is such ha Vn= (0), o n≥k. I ollows
ha belongs o he fini e dimensional Lie subcoalgeb a W=
k−1

n=0
Vn
o L. Hence, one has L=Loc(L) he sum o he fini e dimensional
Lie subcoalgeb as o L. In pa icula , i Lis abelian, one has Vn= (0),
n≥1, and L=L∗.
Mo e gene ally, one sees ha L=Loc(L) iff o each ∈L he
abo e associa ed Lie subcoalgeb a Wo Lis fini e dimensional; in his
case, he e exis s ksuch ha W=
k

n=0
Vn. Ques ion : wha is he class
o all Lie algeb as Lsuch ha L=Loc(L)?
Re e ences
1. W. Michaelis, Lie coalgeb as, Ad ances in Ma h. 38 (1980), 1–54.
2. W. Michaelis, An example o a non-ze o Lie coalgeb a M o
which Loc(M) = (0), J. Pu e Appl. Algeb a 68 (1990), 341–348.
3. W. D. Nichols, The s uc u e o he dual Lie coalgeb a o he
Wi algeb a, J. Pu e Appl. Algeb a 68 (1990), 359–364.
4. W. D. Nichols, On Lie and associa i e duals, J. Pu e Appl. Alge-
b a 87 (1993), 313–320.
5. E. J. Ta , Wi and Vi aso o algeb as as Lie bialgeb as, J. Pu e
Appl. Algeb a 87 (1993), 301–312.

354 B. Dia a
6. E. J. Ta , Algeb aic aspec o linea ly ecu si e sequences, in
“Ad ances in Hop algeb as,” edi ed by J. Be gen, S. Mon gome y,
Ma cel Dekke , New-Yo k, 1994, pp. 299–317.
Keywo ds. Lie coalgeb as
1991 Ma hema ics subjec classifica ions: 16W30
Ma h´ema iques Pu es
Complexe Scien ifique des C´ezeaux
63177 Aubi`e e Cedex
FRANCE
e-mail: [email p o ec ed]cle mon .
P ime a e si´o ebuda el 16 de Ma ¸c de 1995,
da e a e si´o ebuda el 17 de Maig de 1995