ISSN 10645624, Doklady Mathematics, 2012, Vol. 85, No. 1, pp. 109–112. © Pleiades Publishing, Ltd., 2012. Original Russian Text © P.V. Bibikov, V.V. Lychagin, 2012, published in Doklady Akademii Nauk, 2012, Vol. 442, No. 6, pp. 732–735.
MATHEMATICS
Classification of Linear Actions of Algebraic Groups on Spaces of Homogeneous Forms P. V. Bibikov and V. V. Lychagin Presented by Academician V.A. Vasil’ev October 14, 2011 Received October 19, 2011
DOI: 10.1134/S1064562412010383
⺓[A, B, C, D, E], and the invariants A, B, C, D, and E
1. INTRODUCTION This paper studies the orbits of an algebraic group G acting on the space of rational forms in many vari ables by linear changes of coordinates. The main result of the paper consists in finding the field of differential invariants of such an action and obtaining an effective criterion for distinguishing between the orbits of forms with nonzero Hessian. This paper concludes the cycle of papers [1, 2, 8] on classification of orbits of binary and ternary forms. p
Let R n be the space of rational forms of degree n in p variables x1, x2, …, xp over the field ⺓. Consider the p
action of an algebraic group G on R n induced by a linear action of G on the space ⺓p of variables x1, x2, …, xp. In this paper, we describe the orbits of this action. For convenience, we consider pforms of degree n ≤ –1 or n ≥ 2. Note that forms f and ˜f of degree 1 are –1 equivalent if and only if so are the forms f –1 and ˜f of degree –1, and forms f = f1/f2 and ˜f = ˜f 1 /˜f 2 of degree 0 are equivalent if and only if so are the forms λ1 f1 + λ f and λ ˜f 1 + λ ˜f 2 (here, λ and λ are formal vari 2 2
1
2
1
2
ables). Below, we recall known results for the case of the group G = SLp(⺓) acting on the space of polyno mial pforms. In the case of any p and n = 2, the algebra of (poly nomial) Ginvariants is freely generated by one poly nomial, namely, by the Hessian of the form. In the case of p = 4 and n = 3, the algebra of invari ants is ⺓[A, B, C, D, E, F], where the basis invariants have degree 8, 16, 24, 32, 40, 100, respectively, F 2 ∈
Institute of Control Sciences (Automation and Telemechanics), Russian Academy of Sciences, Profsoyuznaya ul. 65, Moscow, 117997 Russia email:
[email protected],
[email protected]
are algebraically independent (see [7]). However, in the framework of the classical theory of invariants, it is impossible to explicitly describe the polynomial algebras of SLp(⺓)invariants for suffi ciently large n and p because of their large homological dimension (see [4]). At the same time, classical theo rems of Hilbert on basis (see [9]) and Rosenlicht (see [10]) are pure existence theorems and do not apply to explicitly describe the invariants of pforms either. This paper presents a method for classifying the Gorbits of regular rational forms of any degree. Sur prisingly, the obtained classification depends neither on the degree n, on the number p of variables, nor even (in some sense) on the group G. As in our preceding papers [1, 2], the main stage in the solution of the problem of classifying the orbits of rational forms is calculating the field of differential invariants of a group G acting on the solution space of the Euler equation p
∑x f
i xi
= nf.
(1)
i=1
After this, we construct an ideal of polynomials (the socalled ideal of dependences) in the fourthorder differential invariants, which completely determines the Gorbit of a regular form. 2. MAIN RESULTS 2.1. The Field of Differential Invariants Let ⺓p be the affine space with coordinates (x1, x2, …, xp). Consider the space Jk = Jk⺓p of kjets of functions with canonical coordinates (x1, x2, …, xp, u, …, uσ) (all necessary definitions can be found in [6]). The Euler differential equation (1) determines the algebraic variety ⊂ J1 specified by the equation p
∑x u
i i
i=1
109
= nu
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BIBIKOV, LYCHAGIN
(here, ui = u0…1…0 – 1 occupies the ith position). Let k denote the (k – 1)th extension (see [6]) of this equa tion to the space Jk of kjets. The action of the group G on the space ⺓p can be canonically extended to an action on all varieties k. Recall that a function J ∈ C∞(k) is called a kth order differential invariant of the action of the group G if J is invariant with respect to the extended action of the group G on the variety k. We consider only invari ants rational in uσ. d d Similarly, a differentiation ∇ = A1 + … + Ap dx 1 dx p (where Ai ∈ C∞(∞)) is said to be invariant if it com mutes with the extended action of the group G. We consider only invariant differentiations with compo nents Ai rational in uσ. Before constructing differential invariants and invariant differentiations, we specify a set of invariant tensors for the action of the group G on the Euler equation. Theorem 1. The horizontal symmetric forms Qk =
∑
σ =k
σ
( dx ) u σ , σ!
given on the space Jk are Ginvariant for all k ≥ 1. We refer to the forms Qk as invariant kforms. Note that these forms do not depend on the group G (it is only important that the group G acts linearly). Now, we describe a basis of invariant differentia tions. Recall that, by an infinite jet, we mean a sequence {θk} of kjets projected onto each other, i.e., such that πk + 1, k(θk + 1) = θk, where πk + 1, k: Jk + 1 → Jk is a natural projection. Geometrically, each (k + 1)jet θk + 1 can be repre sented in the form of a kjet θk and an Rplane L(θk + 1) ⊂ T θk Jk (see [6]). A tangent vector in the space J∞ of infinite jets is a sequence of pairs {(θk, vk)}, where θk ∈ Jk and vk ∈ L(θk + 1), projected onto each other. If this sequence begins with a k0jet, then we say that k0 is the order of the tangent vector. The tangent space T to an infinite jet {θk} of order k0 is the Rplane L( θ k0 + 1 ) at the point θ k0 . It tangent vectors are elements of the induced bundle τ k0 := π *k0 (τ), where τ: T⺓p → ⺓p is the tangent bundle of ⺓p. By a vector field on the space of infinite jets of order k0 we understand a section of the bundle τ k0 . By virtue of the natural projections πk, l: Jk → Jl, any tangent vector of order l is also a tangent vector of each order k > l.
Now, take an infinite jet {θk} of order k0 and let T denote the tangent space of this jet. All further consid erations are in the space T. Note that the invariant kforms Qk can be regarded as symmetric kforms on the space T. We assume that the quadric Q2 is nondegenerate on T. For each tensor v, let v* denote the tensor dual with respect to Q2. By 〈v, w*〉 we denote the convolu tion of tensors. p
Consider the radial differentiation r =
d
∈ T. ∑ x dx i
i=1
i
It is invariant and determines the decomposition of the space T into the direct sum 〈r〉 ⊕ U of subspaces orthogonal with respect to Q2. In what follows, all con siderations are in the subspace U. Take the tensor Q 2* ∈ S2U dual to Q2 and consider the vector ∇1 ∈ U dual to the tensor obtained by pair ing Q3 and Q *2 , that is, ∇ 1 := 〈 Q 3, Q *2 〉 *. Consider also the linear operator D: U → U taking each vector v to the vector dual to the convolution of the cubic Q3 with the symmetric product of ∇1 and v, i.e., defined by D: v 〈 〈 Q 3, ∇ 1〉 , v〉 *. We also set ∇i := Di – 1∇1. Note that the vectors ∇i depend on a point in the space of 3jets and are linearly independent in a Zar iski open subset of the fiber of the projection 3 → 2. Varying the point in the space of 3jets, we obtain the set of differentiations ∇1, ∇2, …, ∇p – 1. Theorem 2. The differentiations r, ∇1, ∇2, …, ∇p – 1 are invariant and form a basis in the space of invariant differentiations. Finally, we are ready to describe the entire field of rational invariants. Note that the values of the kforms Qk on the set of k invariant differentiations are differential invari ants. Let I α := Q 3 ( ∇ α1, ∇ α2, ∇ α3 ), where α = (α1, α2, α3) is an ordered set of indices, i.e., consider the coefficients1 of the form Q3 in the “invari ant basis” {r*, ∇ *1 , ∇ *2 , …, ∇ *p – 1 }. Theorem 3. The field of differential invariants of an action of an algebraic group G on the manifold ∞ is gen erated by the differential invariants Ji of order at most 2, by the differential invariants Iα of order 3, and by the invariant differentiations ∇1, ∇2, …, ∇p – 1. Moreover, 1 It can be proved that all coefficients before r* vanish, and there
fore, the differentiation r does not participate in the definition of the invariants Iα. DOKLADY MATHEMATICS
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CLASSIFICATION OF LINEAR ACTIONS OF ALGEBRAIC GROUPS
this field is algebraically generated by the invariants Ji and by derivatives of the form ∇σIα. It separates the Gorbits of jets of maximal dimension. It follows from this theorem that the actions of groups differ only in invariants of order at most 2, while the main “infinite part” is the same and does not depend on the explicit form of the group G. 2.2. GOrbits of Homogeneous Forms In this section, we apply Theorem 3 to explicitly describe the Gorbits of homogeneous pforms with nonzero Hessian. For this purpose, consider the fourthorder differential invariants J i, I α, I α, k := ∇ k I α . Their restrictions to the graph L f ⊂ 4 of a form f with nonzero Hessian (this requirement is necessary, because otherwise, the denominators of some invari ants vanish) are homogeneous rational functions in the variables x1, x2, …, xp and determine the rational mapping 4
p
N
ρf : ⺓ → ⺓ ,
4
4
ρ f ( a ) = ( J 1 ( [ f ] a ), …, I 111 ( [ f ] a ), … )
(where N is the number of the chosen invariants). Thus, there are algebraic dependences between these restrictions. We denote the set of such dependences by f and the image of the mapping ρf by Φf. We refer to f as the ideal of dependences of the pform f. The main theorem of this paper is as follows. Theorem 4. 1. Forms f and ˜f with nonzero Hessian are Gequivalent if and only if Φf = Φ f˜ .
In this section, we apply the method for classifying the Gorbits of pforms described above to the groups (i) G = SLp(⺓); (ii) G = GLp(⺓); (iii) G = SOp(⺓). We start with the case G = SLp(⺓). To construct the field of differential invariants, we must calculate the differential invariants of orders 0, 1, and 2 of the group SLp(⺓). It is well known that there exists a unique invariant U = u of order 0, there are no nontrivial differential invariants of order 1, and there exists a unique nontriv ial differential invariant H of order 2, which is the Hes sian of the quadric Q2 (see [7]). Vol. 85
Now, to describe the SLp(⺓)orbits of homoge neous pforms, it suffices to apply Theorem 4. Consider the case G = GLp(⺓). We assume that the subgroup SLp(⺓) ⊂ GLp(⺓) acts by linear changes of coordinates, and the center ⺓* ⊂ GLp(⺓) acts by homotheties f λf, where f ∈ R n and λ ∈ ⺓*. p
As is known (see [7]), there are no nontrivial invari ants of orders 0 and 1, and there exists a unique invari ˜ := H/up of order 2. ant H ˜ k := Moreover, in this case, invariant forms are Q Qk/u rather than Qk. Therefore, applying the theorems ˜ k , we of Sections 1 and 2 to the invariant kforms Q obtain the field of differential invariants of the action of the group GLp(⺓). Now, to describe the GLp(⺓)orbits of homoge neous pforms, it suffices to apply Theorem 4. Note that the classification of the GLp(⺓)orbits of polynomial pforms of degree n is equivalent to the projective classification of irreducible algebraic pro jective hypersurfaces of degree n in ⺓P p. Finally, consider the case G = SOp(⺓) (the case p = 3 has been considered earlier in [3]). It is easy to prove that there exists a unique invari n
∑x
2 i
of order –1, a unique invariant U = u of
i=1
n
order 0, and a unique invariant N =
∑u
2 i
(“gradi
i=1
ent”). There are precisely p differential invariants K1, K2, …, Kp of order 2, which are the coefficients in the characteristic polynomial χ Q2 of the quadric Q2 (see [5]).
2.3. Applications
DOKLADY MATHEMATICS
Thus, the field of differential invariants of the group SLp(⺓) is generated by the invariant U of order 0, the invariant H of order 2, the canonical invariants Iα, and the invariant differentiations ∇1, ∇2, …, ∇p – 1.
ant d =
2. Forms f and ˜f with nonzero Hessian are Gequiv alent if and only if their ideals of dependences coincide: f = f˜ .
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Thus, the field of differential invariants of the group SOp(⺓) is generated by the invariants d, U, N, the invariants K1, K2, …, Kp of order 2, the canonical invariants Iα, and the invariant differentiations ∇1, ∇2, …, ∇p – 1. Now, to describe the SOp(⺓)orbits of homoge neous pforms, it suffices to apply Theorem 4. ACKNOWLEDGMENTS P.V. Bibikov acknowledges the support of the Simons Foundation and of the program for support of young can didates of sciences (project no. MK32.2011.1).
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REFERENCES 1. P. Bibikov and V. Lychagin, Dokl. Math. 82, 915–917 (2010). 2. P. Bibikov and V. Lychagin, Dokl. Math. 84, 482–484 (2011). 3. P. Bibikov, Izv. PGPU, No. 26, 36–42 (2011). 4. V. Popov, Izv. Akad. Nauk SSSR, Ser. Mat. 47 (3), 310– 334 (1983).
5. E. Vinberg and V. Popov, Theory of Invariants (VINITI, Moscow, 1989) [in Russian]. 6. D. Alekseevskii, A. Vinogradov, and V. Lychagin, Basic Ideas and Notions of Differential Geometry (VINITI, Moscow, 1988) [in Russian]. 7. J. Dixmier, Soc. Math. Fr. 43, 39–64 (1990). 8. B. Kruglikov and V. Lychagin, Int. J. Geom. Methods Mod. Phys. 3 (5–6), 1131–1165 (2006). 9. D. Hilbert, Math. Ann. 36, 473–534 (1890). 10. M. Rosenlicht, Am. J. Math. 78, 401–443 (1956).
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