On the definition of the dual lie coalgebra of a lie algebra

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