2000 Mathematics Subject Classification: 13N15, 13A50, 16W25.We reduce the Nowicki conjecture on Weitzenböck derivations of polynomial algebras to a well known problem of classical invariant theory

Bedratyuk, Leonid

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Serdica Math. J. 35 (2009), 311–316A NOTE ABOUT THE NOWICKI CONJECTUREON WEITZENBO¨CK DERIVATIONSLeonid BedratyukCommunicated by V. DrenskyAbstract. We reduce the Nowicki conjecture on Weitzenbo¨ck derivationsof polynomial algebras to a well known problem of classical invariant theory.1. Let K be a field of characteristic 0. A linear locally nilpotent derivationD of the polynomial algebra K[Z] = K[z1, z2, . . . , zm] is called a Weitzenbo¨ckderivation. It is well known that the kernelkerD := {f ∈ K[Z] | D(f) = 0}of the linear locally nilpotent derivation D is a finitely generated algebra, see[13, 11, 12].LetK[X,Y ] = K[x1, x2, . . . , xn, y1, y2, . . . , yn] be the polynomialK-algebrain 2n variables. Consider the following Weitzenbo¨ck derivation D1 of K[X,Y ]:D1(xi) = 0,D1(yi) = xi, i = 1, 2, . . . , n.2000 Mathematics Subject Classification: 13N15, 13A50, 16W25.Key words: Classical invariant theory, covariants of binary form, derivations.312 Leonid BedratyukIn [8] Nowicki conjectured that kerD1 is generated by the elements x1, x2, . . . , xnand the determinants ∣∣∣∣ xi xjyi yj∣∣∣∣ , 1 ≤ i < j ≤ n.The conjecture was confirmed by several authors, see [6, 1, 7].In this note we show that the Nowicki conjecture is equivalent to a wellknown problem of classical invariant theory, namely, to the problem to describethe algebra of joint covariants of n linear binary forms. Using the same idea wepresent an explicit set of generators of the kernel of the derivation D2 ofK[X,Y,Z] = K[x1, . . . , xn, y1, . . . , yn, z1, . . . , zn]defined byD2(xi) = 0,D2(yi) = xi,D2(zi) = yi, i = 1, . . . , n.2. It is well known that there is a one-to-one correspondence between theGa-actions on an affine algebraic variety V and the locally nilpotent K-derivationson its algebra of polynomial functions. Let us identify the algebra K[X,Y ] withthe algebra O[K2n] of polynomial functions of the algebraic variety K2n. Then,the kernel of the derivation D1 coincides with the invariant ring of the inducedvia exp(tD1) action:kerD1 = O[K2n]Ga = K[X,Y ]Ga .Now, letV1 := {αX + βY | α, β ∈ K}be the vector K-space of linear binary forms endowed with the natural action ofthe group SL2. Consider the induced action of the group SL2 on the algebra ofpolynomial functions O[nV1 ⊕K2] on the vector space nV1 ⊕K2, wherenV1 := V1 ⊕ V1 ⊕ . . . ⊕ V1︸ ︷︷ ︸n times.LetU2 ={(1 λ0 1) ∣∣∣ λ ∈ K}be the maximal unipotent subgroup of the group SL2. The application of theGrosshans principle, see [5, 10], givesO[nV1 ⊕K2]SL2 ∼= O[nV1]U2.Nowicki conjecture on Weitzenbo¨ck derivations 313Thus,O[nV1 ⊕K2] sl2 ∼= O[nV1]u2 .Since U2 ∼= (K,+), it follows thatkerD1 ∼= O[nV1 ⊕K2] sl2 ∼= O[nV1]u2 .In the language of classical invariant theory the algebraC1 := O[nV1 ⊕ K2] sl2 is called the algebra of joint covariants of n linear binaryforms and the algebra S1 := O[nV1]u1 is called the algebra of joint semi-invariantsof n linear binary forms. Algebras of joint covariants of binary forms were anobject of research in the classical invariant theory of the 19th century.3. Let us consider the set of n linear binary forms fi = xiX + yiY,i = 1, . . . , n. Then any element of O[nV1⊕K2] can be considered as a polynomialfrom K[X,Y,X ,Y]. Gordan’s famous theorem, see [3, 2], implies:Theorem 1 (A weak form of Gordan’s theorem). If T is a subalgebra ofC1 with the property that (fi, z)r ∈ T whenever r ∈ N, z ∈ T , then T = C1.Here (u, v)r denotes the r-transvectants of the binary formsu, v ∈ K[X,Y,X ,Y]:(u, v)r :=r∑i=0(−1)i(ri)∂ru∂X r−i∂Y i∂rv∂X i∂Yr−i.Observe, that (u, v)0 = u v and (u, v)1 is exactly the Jacobian J(u, v) ofthe forms u, v. The above theorem yields:Theorem 2. The algebra of joint covariants C1 of n linear binary formsfi, i = 1, . . . , n, is generated by the forms f1, f2, . . . , fn and their jacobiansJ(fi, fj), 1 ≤ i < j ≤ n.P r o o f. All forms fi, i = 1, . . . , n, belong to the algebra of covariants C1.By direct calculations we get(fi, fj)1 = J(fi, fj) =∣∣∣∣∣∣∣∣∣∂fi∂X∂fi∂Y∂fj∂X∂fi∂Y∣∣∣∣∣∣∣∣∣=∣∣∣∣ xi xjyi yj∣∣∣∣ ,and(fi, fj)r = 0 for r > 1.314 Leonid BedratyukLet us consider the subalgebra T of C1 generated by the linear forms f1, f2, . . . ,fn and theirs jacobians J(fi, fj), i < j. Since (fi, J(fj , fk))r = 0 for all r ≥ 1, itfollows that T = C1. Let us show that the result is equivalent to the Nowicki conjecture:Identify the algebra of semi-invariants S1 with kerD1. The isomorphismτ : C1 → S1 takes each homogeneous covariant of degree m (with respect to thevariables X ,Y) to its coefficient of Xm. The proof proceeds in the same manneras the proof in the case of a unique binary form, see [9, Proposition 9.45].Thus, the following statement holds:Theorem 3. The algebra of joint semi-invariants S1 = kerD1 of n linearbinary forms fi, i = 1, . . . , n, is generated by the elements x1, x2, . . . , xn and thedeterminants ∣∣∣∣ xi xjyi yj∣∣∣∣ , 1 ≤ i < j ≤ n.P r o o f. The algebra S1 is generated by the images of the generatingelements of the algebra C1 under the homomorphism τ . We have τ(fi) = xi andτ(J(fi, fj)) =∣∣∣∣ xi xjyi yj∣∣∣∣ . Theorem 3 is exactly the Nowicki conjecture.4. Other ways to prove Theorem 3 were suggested by Dersken and Pa-nyushev, see the comments in [1]. Taking into account K2 ∼=sl2 V1 we getkerD1 ∼= O[nV1 ⊕K2] sl2 ∼= O[(n+ 1)V1]sl2 .But then the invariant algebra O[(n+1)V1]sl2 is well known because of the Firstfundamental theorem of invariant theory for SL2, see [14].5. A natural generalization of the above problem looks as follows.LetK[X,Y,Z] = K[x1, . . . , xn, y1, . . . , yn, z1, . . . , zn]be the polynomial K-algebra in 3n variables. Consider the following derivationD2 of the algebra K[X,Y,Z]:D2(xi) = 0, D2(yi) = xi, D2(zi) = yi, i = 1, 2, . . . , n.Nowicki conjecture on Weitzenbo¨ck derivations 315The following theorem holds.Theorem 4. The kernel of the derivation D2 is generated by the elementsof the following types:1. x1, x2, . . . , xn,2. J1,2, J1,3, . . . , Jn−1,n,3. H1,2,H1,3, . . . ,Hn−1,n,4. ∆1,2,3,∆1,2,4, . . . ,∆n−2,n−1,n,whereJi,j :=∣∣∣∣ xi xjyi yj∣∣∣∣ , 1 ≤ i < j ≤ n,Hi,j = xizj − yiyj + zixj , 1 ≤ i ≤ j ≤ n,and∆i,j,k :=∣∣∣∣∣∣xi xj xkyi yj ykzi zj zk∣∣∣∣∣∣ , 1 ≤ i < j < k ≤ n.The proof follows from the description of the generating elements of thealgebra of covariants for n quadratic binary forms, see [4, page 162].6. Any Weitzenbo¨ck derivation of the polynomial algebra is completelydetermined by its Jordan normal form. Denote by Dk the Weitzenbo¨ck derivationwith Jordan normal form which consists of n Jordan blocks of size k + 1.Problem. Find a generating set of kerDk.REFERENCES[1] V. Drensky, L. Makar-Limanov. The conjecture of Nowicki onWeitzenbo¨ck derivations of polynomial algebras. J. Algebra Appl. 8, 1 (2009)41–51.[2] O. E. Glenn. Treatise on Theory of Invariants. Boston, 1915.[3] P. Gordan. Vorlesungen u¨ber Invariantentheorie. Teubner, Leipzig, 1885.Reprinted by Chelsea Publishing Company, New York, 1987.316 Leonid Bedratyuk[4] J. Grace, A. Young. The Algebra of Invariants. Cambrige Univ. Press,1903.[5] F. Grosshans. Observable groups and Hilbert’s fourteenth problem. Amer.J. Math. 95 (1973), 229–253.[6] J. Khoury. Locally Nilpotent Derivations and Their Rings of Constants,Ph.D. Thesis, Univ. Ottawa, 2004.[7] S. Kuroda. A simple proof of Nowicki’s conjecture on the kernel of anelementary derivation. Tokyo J. Math. 32, 1 (2009), 247–251.[8] A. Nowicki. Polynomial Derivations and their Rings of Constants.Uniwersytet Mikolaja Kopernika, Torun, 1994 [Available at:http://www-users.mat.uni.torun.pl/~anow/polder.html.][9] P. Olver. Classical Invariant Theory. Cambridge University Press, 1999.[10] K. Pommerening. Invariants of unipotent groups. – A survey. In: InvariantTheory, Symp. West Chester/Pa, 1985, Lect. Notes Math. vol. 1278, 1987,8–17.[11] C. S. Seshadri. On a theorem of Weitzenbo¨ck in invariant theory. J. Math.Kyoto Univ. 1 (1962), 403–409.[12] A. Tyc. An elementary proof of the Weitzenbo¨ck theorem. Colloq. Math.78 (1998), 123–132.[13] R. Weitzenbo¨ck. U¨ber die Invarianten von linearen Gruppen. Acta Math.58 (1932), 231–293.[14] H. Weyl. The Classical Groups. Their Invariants and Representations.Princeton University Press, 1997.Khmelnytsky National University11, Instytuts’ka st.29016 Khmelnytsky, Ukrainee-mail: leonid.uk@gmail.com Received October 3, 2009

A Note About the Nowicki Conjecture on Weitzenböck Derivations

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A Note About the Nowicki Conjecture on Weitzenböck Derivations

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