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

Publicacions Matem`atiques, Vol 39 (1995), 349–354. ON THE DEFINITION OF THE DUAL LIE COALGEBRA OF A LIE ALGEBRA Bertin Diarra Abstract Let Lbe a Lie algebra over a field K. The dual Lie coalgebra L◦ of Lhas been defined by W. Michaelis to be the sum of all good subspaces Vof the dual space L∗of L:Vis good if tm(V)⊂ V⊗V, where mis the multiplication of L. We show that L◦= tm−1(L∗⊗L∗) as in the associative case. Let Lbe a Lie algebra over the field Kwith multiplication m:L⊗L→ L: i.e. mis a linear map and setting m(x⊗y)=[x, y], one has (1) [x, x]=0 (2) [x, [y,z]]+[z,[x, y]] + [y, [z,x]] = 0. Let L∗be the dual vector space of Land tm:L∗→(L⊗L)∗be the transpose of m. We identify L∗⊗L∗with a subspace of (L⊗L)∗and we set L=tm−1(L∗⊗L∗). Fix f∈L∗and consider the linear map γf:L→L∗defined by (3) γf(x),y=f,[x, y]=tm(f),x⊗y,x,y∈L. Setting, as usual, adx(y)=[x, y], one has γf(x)=t(adx)(f). Sometimes, we shall write γf(x)=x·f. 350 B. Diarra Lemma 1. For f∈L∗, the following statements are equivalent (i) f∈L (ii) The linear map γf:L→L∗is of finite rank. Proof: The equivalence follows readily from (3). Moreover, since L∗⊗L∗can be identified with the space of the linear maps of Linto L∗of finite rank, via ι:L∗⊗L∗→Hom(L, L∗)by setting ι(f⊗g)(x)=f(x)g, one has f∈Liff γf= n  j=1 fj⊗gjiff tm(f)= n  j=1 fj⊗gj. Lemma 2. For f∈Land x∈L, one has, γf(x)=x·f∈L. Moreover γf(L)is a vector subspace of Lof finite dimension. Proof: If f∈L, then for x,y∈L, one has f,[x, y]=γf(x),y= n  j=1 fj,xgj,y. However, (2) can be written [x, [y,z]] = −[y, [z,x]] − [z,[x, y]]. Therefore, one has γx·f(y),z=x·f,[y,z] =f,[x, [y,z]] =−f,[y,[z,x]]−f,[z,[x, y]] =− n  j=1 fj,ygj,[z,x]− n  j=1 fj,zgj,[x, y] = n  j=1 fj,yx·gj,z− n  j=1 x·gj,yfj,z. Hence γx·f(y)= n  j=1 fj,yx·gj− n  j=1 x·gj,yfj. It follows that γx·f is of finite rank, that is x·f=γf(x)∈L. Then, it is clear that γf(L) is a vector subspace of Land finite dimensional. Dual Lie coalgebra of a Lie algebra 351 Note. One deduces from the above proof that if f∈Land tm(f)= n  j=1 fj⊗gjthen for x∈L, one has tm(x·f)= n  j=1 fj⊗(x·gj)− n  j=1 (x·gj)⊗fj. Theorem 1. Lis a good subspace of L∗i.e. tm(L)⊂L⊗L. Moreover, one has L=L◦. Proof: Let f∈Land let (gj)1≤j≤nbe a base of γf(L)⊂L. One has gj=xj·fand for any x∈L,γf(x)= n  j=1 fj(x)gj; hence tm(f)= n  j=1 fj⊗gj,fj∈L∗. However m=−m◦τ(skew-symmetry), therefore, one has tm=−tτ◦tmand tm(f)= n  j=1 fj⊗gj=−tτ( n  j=1 fj⊗gj)= − n  j=1 gj⊗fj. Since (gj)1≤j≤nis free in L∗, there exists for 1 ≤≤n y∈Lsuch that gj,y =δj. Hence (y⊗1L∗)(tm(f)) = n  j=1 fj,y gj= − n  j=1 gj,y fj=−f, that is f=− n  j=1 fj,y gj∈γf(L). It follows that tm(f)= n  j=1 fj⊗gj∈γf(L)⊗γf(L)⊂L⊗Land L⊂L◦. On the other hand, it is clear that any good subspace Vof L∗is contained in L, therefore L◦⊂L. We have proved that L=L◦. Note. If f∈Land if (xj·f)1≤j≤nis a base of γf(L), one has tm(f)= n  j=1 (yj·f)⊗(xj·f) where, for 1 ≤j≤n,yjis such that f,[x,y j]=δj. Furthermore, if x∈L, one has (4) tm(x·f)= n  j=1 (yj·f)⊗[x·(xj·f)] − n  j=1 [x·(xj·f)] ⊗(yj·f). 352 B. Diarra Remark. Put ∆ = tm|L:L→L⊗L. Following W. Michaelis [1], (see also [2], [3], [4], [5] and [6]) one obtains a Lie coalgebra (L,∆), that is : (5) ∆ = −τ◦∆ if the characteristic of Kis different from 2 and Im ∆ ⊂Im(1L−τ) otherwise [τ(f⊗g)=g⊗f]. (6) (id3+σ+σ2)◦(1L⊗∆) ◦∆=0 where σ(f⊗g⊗h)=h⊗f⊗g. This follows from (1) and (2). Notice that (2) is equivalent to m◦ (1L⊗m)◦(id3+ρ+ρ2) = 0 where ρ(x⊗y⊗z)=z⊗x⊗yand one has tρ2|L=σ. For A⊂L, let span(A) be the vector subspace of Lspaned by A. Theorem 2. Let f∈L. Put V0=K·f V1=γf(L)={x1·f,x1∈L} V2= span{x2·f1,x 2∈L, f1∈V1} ................................................... Vn= span{xn·fn−1,x n∈L, fn−1∈Vn−1} ............................................................... Then W= n≥0 Vnis a Lie subcoalgebra of Land is the smallest Lie subcoalgebra of Lthat contains f. Proof: We have seen that if f∈L, then ∆(f)= n  j=1 (yj·f)⊗(xj·f). It follows that ∆(V0)⊂V1⊗V1. Furthermore V1⊂L, and by induction one has Vn⊂L. On the other hand, if xn∈L,fn−1∈Vn−1,n≥1, one has by (4) ∆(xn·fn−1)= m  j=1 (ynj ·fn−1)⊗[xn·(xnj ·fn−1)] − m  j=1 [xn·(xnj ·fn−1)] ⊗(ynj ·fn−1)∈Vn⊗Vn+1 +Vn+1 ⊗Vn. Therefore, ∆(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 subcoalgebra of L. Dual Lie coalgebra of a Lie algebra 353 Let Vbe a Lie subcoalgebra of L. For 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. Therefore, if Vcontains f, one has V0⊂Vand V1=γf(L)⊂V.It is readily seen by induction that Vn⊂Vfor all n≥0. It follows that W= n≥0 Vnis contained in V. Note. (i) One can prove, by induction, that the above Vn,n≥0, are finite dimensional. (ii) One has for n≥1, Vn= span{tad x1◦tad x2◦...◦tad xn(f), x1,... ,x n∈L}. Therefore, if Lis nilpotent of class k, then for any f∈L, the associated sequence of subspaces (Vn)n≥0is such that Vn= (0), for n≥k. It follows that fbelongs to the finite dimensional Lie subcoalgebra W= k−1  n=0 Vn of L. Hence, one has L=Loc(L) the sum of the finite dimensional Lie subcoalgebras of L. In particular, if Lis abelian, one has Vn= (0), n≥1, and L=L∗. More generally, one sees that L=Loc(L) iff for each f∈Lthe above associated Lie subcoalgebra Wof Lis finite dimensional; in this case, there exists ksuch that W= k  n=0 Vn. Question : what is the class of all Lie algebras Lsuch that L=Loc(L)? References 1. W. Michaelis, Lie coalgebras, Advances in Math. 38 (1980), 1–54. 2. W. Michaelis, An example of a non-zero Lie coalgebra Mfor which Loc(M) = (0), J. Pure Appl. Algebra 68 (1990), 341–348. 3. W. D. Nichols, The structure of the dual Lie coalgebra of the Witt algebra, J. Pure Appl. Algebra 68 (1990), 359–364. 4. W. D. Nichols, On Lie and associative duals, J. Pure Appl. Algebra 87 (1993), 313–320. 5. E. J. Taft, Witt and Virasoro algebras as Lie bialgebras, J. Pure Appl. Algebra 87 (1993), 301–312. 354 B. Diarra 6. E. J. Taft, Algebraic aspect of linearly recursive sequences, in “Advances in Hopf algebras,” edited by J. Bergen, S. Montgomery, Marcel Dekker, New-York, 1994, pp. 299–317. Keywords. Lie coalgebras 1991 Mathematics subject classifications: 16W30 Math´ematiques Pures Complexe Scientifique des C´ezeaux 63177 Aubi`ere Cedex FRANCE e-mail: [email protected]clermont.fr Primera versi´o rebuda el 16 de Mar¸c de 1995, darrera versi´o rebuda el 17 de Maig de 1995