On maximality of some solvable and locally nilpotent subalgebras of the Lie algebra $W_n(K)$

Let $K$ be an algebraically closed field of characteristic zero, $P_n=K[x_1, ..., x_n]$ the polynomial ring, and $W_n(K)$ the Lie algebra of all $K$-derivations on $P_n$. One of the most important subalgebras of $W_n(K)$ is the triangular subalgebra $u_n(K) = P_0\partial_1+\cdots+P_{n-1}\partial_n$, where $\partial_i:=\partial/\partial x_i$ are partial derivatives on $P_n$. This subalgebra consists of locally nilpotent derivations on $P_n.$ Such derivations define automorphisms of the ring $P_n$ and were studied by many authors. The subalgebra $u_n(K) $ is contained in another interesting subalgebra $s_n(K)=(P_0+x_1P_0)\partial_1+\cdots +(P_{n-1}+x_nP_{n-1})\partial_n,$ which is solvable of the derived length $2n$ that is the maximum derived length of solvable subalgebras of $W_n(K).$ It is proved that $u_n(K)$ is a maximal locally nilpotent subalgebra and $s_n(K)$ is a maximal solvable subalgebra of the Lie algebra $W_n(K).$


Introduction
vet K e n lgerilly losed (eld of hrteristi zero nd P n = K[x 1 , . . ., x n ] the polynomil ring in n vrilesF ell tht KEliner mp D : P n → P n is lled KEderivtion @or simply derivtion if the (eld K is (xedA if it stis(es the veiniz ruleX D(f g) = D(f )g + f D(g) for ny f, g ∈ P n F por ny f 1 , . . ., f n ∈ P n there exists unique , where ∂ i := ∂/∂x i re prtil derivtives on P n F he vetor spe W n (K) of ll KEderivtions on P n is vie lger over the (eld K with respet to the vie rket xilpotentD lolly nilpotent nd solvle sulgers of W n (K) were studiE ed y mny uthorsD strted from S @seeD for exmpleD PD QD TAF yne of the most importnt sulgers of W n (K) is the tringulr vie lger whih onsists of lolly nilpotent derivtions on K[x 1 , . . ., x n ]. his vie lger is lolly nilpotent ut not nilpotentD its struture nd properties were studied in PF e onsider its emedding in W n (K) nd prove tht u n (K) is mximl lolly nilpotent sulger of W n (K) @heorem IAF enother mxiE mlity property of u n (K) ws onsidered in VD where it ws proved tht u n (K) is mximl sulger ontined in the set of lolly nilpotent derivtions on P n @note tht this set is not vie sulger of W n (K)AF sn TD it ws proved tht the derived length of solvle sulgers in W n (K) does not exeed 2nF he known exmple of solvle sulgers tht rehes this ound ws pointed out in UD this is the sulger st is ler tht the sulger u n (K) is properly ontined in s n (K)F he sulger s n (K) hs lso mximlity propertyX we prove tht s n (K) is mximl solvle sulger of W n (K) @heorem PAF xote tht s n (K) ppers in nturl wy while studying vie lgers of vetor (elds on C n @see UAF sn generlD mximl sulgers of the vie lger W n (K) re not desriedD ut some types of suh sulgers re known @seeD for exmpleD IAF xote tht the struture of mximl sulgers of semisimple vie lgers ws desried in RF e use stndrd nottionsF ell tht derivtion D ∈ W n (K) is lled lolly nilpotent if for ny f ∈ P n there exists positive integer k PV the mximum of @totlA degrees of its omponentsD iFeF polynomils f 1 , . . ., f n .vet f = f (x 1 , . . ., x n ) ∈ P n e polynomilF e sy tht f hs n index s if

Maximality of u n (K)
e need some tehnil lemms to prove heorem ID the min result of this setionF vemm P seems to e known ut hving no ext referenes we point out its proof for ompletenessF Lemma 1 @TD vemm IA. vet where sf the polynomil t 1 (x 1 , . . .x i−1 , x i+1 , . . ., x n ) is nononstnt then y the (rst prt of this lemm there exist nonnegtive integers β 1 , . . .
PW hF sF ipswyD wF F hyyD uF eF eu henoting where Lemma 3. sf there exists lolly nilpotent sulger S of the vie lger W n (K) tht properly ontins u n (K)D then there exists @nonzeroA liner deriE vtion Proof.uppose the sttement of the lemm is flse nd the set S \ u n (K) does not ontin ny nonzero liner derivtionF vet us hoose derivtion D ∈ S\u n (K) of minimum degree nd write it in the form . he ltter ontrdits the proven oveF oD f i = f i0 + λ ij x j for some λ ij ∈ K. epeting these onsidertions for every j ≥ i we see tht f i n e hosen in the form pirstlyD let us prove tht every liner derivtion D ∈ S \ u n (K) is digonlD iFeF tht the mtrix (λ ij ) n i,j=1 is digonlF vet it e not the se nd hoose ny liner @nonEdigonlA derivtion eigenvetor for the liner opertor ad D 0 with the eigenvlue λ ij .ine D is nonEdigonl @y our ssumptionA there exists nonzero oe0ient λ ij , i < j. he ltter is impossile euse the sulger S is lolly nilpotentF he otined ontrdition shows tht ll liner derivtions from S \ u n (K) re digonlF ke ny liner derivtion he ltter mens tht x i ∂ j is n eigenvetor for the liner opertor ad D with the @nonzeroA eigenvlue µ i −µ j D whih is impossile euse S is lolly nilpotent sulger of W n (K)F hereforeD we hve tht D = µE n for some ⊂ SF he ltter is impossile s it ws mentioned oveF he otined ontrdition shows tht S = u n (K) nd u n (K) is mximl lolly nilpotent sulger of the vie lger W n (K)F

Maximality of s n (K)
ell tht we denote y P i = K[x 1 , . . ., x i ] the polynomil ring over K. e lso denote for onveniene P 0 = K. st is esy to see tht the KEsuspe his sulger is solvle of the derived length 2n.ome properties of s n (K) were pointed out in UF ine the derived length of solvle sulgers of W n (K) does not exeed 2n @see TD UAD the sulger s n (K) hs the mximum possile derived lengthF rere we prove tht s n (K) is mximl solvle sulger of W n (K) @heorem PAF vet D ∈ W n (K) e derivtion of the form he rule of ommuttion of genertors T 1 , T 2 , T 3 shows tht ϕ n e extended to homomorphism of the vie lger Proof.uppose to the ontrry tht there exists solvle sulger S ⊂ W n (K) suh tht s n (K) is properly ontined in S. henote y k the smllest index of derivtions from the set S \ s n (K) nd onsider the set D k of ll derivtions D ∈ S \ s n (K) tht hve the index k.vet us hoose derivtion D ∈ D k in suh wy tht its @nonzeroA polynomil oe0ient f k @y the prtil derivtive ∂ k A hs the smllest index s.hen we hve where where e investigte the possile two sesX s > k nd s = k.
gse IF vet us egin with the se s > k.
vet us expnd the polynomil f k ∈ P s from the derivtion D @written in the form @QFIAA in powers of x s : for some l ≥ 1D where g i ∈ P s−1 , g l = 0. hen the produt . .] ∈ S n e written in the form , where K * is the group of units of the (eld K. epplying this opertor to the derivtion D 0 we otin derivtion D 1 of the form gonsider the suse s − 1 > k. hen it holds king into ount the reltions @QFPA nd @QFQAD we otin , nd we get ontrditionD sine the polynomil λx s−1 hs the index less thn s. QQ hF sF ipswyD wF F hyyD uF eF eu xow let s − 1 = k.hen ∂ k (x s ) = 0, so we get ine ∂ i (x s ) = 0 for ll i = 1, . . ., k − 1 one n esily show @using the reltiE on @QFRAA tht we see y vemm R tht S is nonEsolvleF he ltter ontrdits the hoie of S nd this ontrdition shows tht the se s > k is impossileF gse PF xow let us onsider the se s = k.es shown oveD in this se where g l = 0, l ≥ 2, g i ∈ P k−1 , i = 1, . . ., l.