Square-Zero Basis of Matrix Lie Algebras

A method is obtained to compute the maximum number of functionally independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a base of square-zero matrices. Some applications are also given.


A class of Lie algebras
Square-zero matrices have been dealt with in several settings and with different purposes; for example, see [2], [3], [4], [8], [10], [13], among other authors. Below, we consider such matrices in connection with the following question: (*) Let F be a field. We try to find out whether a given Lie subalgebra g in gl(n, F) admits a basis B (as a vector space over F) such that the square of any matrix in B is zero.
Lemma 1.1. Let G ⊆ GL(n, F) be a linear algebraic group with associated Lie algebra g. If U is a square-zero matrix in the Lie subalgebra g ⊂ gl(n, F), then I + tU belongs to G, ∀t ∈ F, where I ∈ GL(V ) denotes the identity map.
Proof. If U ∈ g is a square-zero matrix, then H = {I + tU : t ∈ F} is a linear algebraic group of dimension 1 with Lie algebra h = {tU : t ∈ F}, and by virtue of the assumption, we have g∩h = h. Hence dim(G∩H) = dim H = 1, so that H = G ∩ H, or equivalently H ⊆ G.
Notation 1.2. Notations and elementary properties of algebraic sets and groups have been taken from Fogarty's book [6]. The Lie algebra g of a linear algebraic group G is identified to the Lie algebra of left-invariants derivations (cf. [6, 3.17 If A = (a ij ) n i,j=1 ∈ gl(n, F), then the corresponding invariant derivation is given by The importance of (*) lies in the following result: Theorem 1.4. Let G be a linear algebraic group, let ρ : G → GL(n, F) be a linear representation of G, and let ρ * : g → gl(n, F) be the homomorphism of Lie algebras induced by ρ. If V = F n and g satisfies the property (*), then every G-invariant function I ∈ F[V * ] is a common first-integral of the system of derivations ρ * (X), ∀X ∈ g. Hence, the number of algebraically independent G-invariant functions in F[V * ] is upper bounded by the difference n 2 − r, where r is the generic rank of the F[V * ]-module M spanned by all the derivations ρ * (X), ∀X ∈ g; i.e., r is the dimension of the F(V * )-vector Proof. Let B be basis for g by fulfilling the condition in (*). By virtue of Lemma 1.1, the matrix I + tB belongs to G and we have I ((I + tB) · v) = I(v), for all t ∈ F, B ∈ B, and by taking derivatives at t = 0, we deduce that ρ * (X)(I) = 0, ∀X ∈ g, because the map spanned by the invariant vector fields ρ * (B i ), 1 ≤ i ≤ m, and the differential dI ∈ Ω F (F[V * ]) of every invariant function I verifies dI(X) = 0, ∀X ∈ M, and we can conclude.
In classical invariant theory over complex numbers a method for computing the maximum number of algebraically independent invariants consists in solving the linear equations arising from the system of first integrals of vector fields ρ * (X), X ∈ g; e.g., see [12,Theorem 4.5.2]. Theorem 1.4 extends this procedure to a class of linear representations in positive characteristic.
It would also be interesting to adapt the algorithms given in [5] to the linear representations of a linear algebraic group whose Lie algebra satisfies the condition (*) in positive characteristic. Remark 1.5. As ρ * : g → gl(n, F) is a homomorphism of Lie algebras, M is an involutive submodule in Der F[V * ]. In the real or complex cases, Frobenius theorem implies that the maximum number of algebraically independent firstintegral functions of M is n 2 − r exactly, but in general the upper bound n 2 − r is not necessarily reached as several of these first-integral functions may be fractional or even transcendental functions. Nevertheless, we have Proof. According to [7,Théorème 1], in the two cases of the statement above where the polynomials p 1 , . . . , p m are algebraically independent. Hence their differentials dp 1 , . . . , dp m form a basis of the dual module to Der F (F[V * ]) G by virtue of [9, VIII. Proposition 5.5], and we thus obtain m = n 2 − r.
we deduce that the basic invariant is the function I 1 : O → F defined on the Zariski open subset O of non-degenerate metrics as follows: . Nevertheless, the result depends strongly on the linear representation being considered. For example, if we consider the natural representation of GL(2, , which is globally defined, and, in this case, we have is considered, then, besides I 1 , there exists another globally defined invariant, namely the discriminant function, i.e., I 2 (v, s) = zu − t 2 . Hence I 1 I 2 is also globally defined and we have F[V * ] SL(2,F) = F[I 1 I 2 , I 2 ]. Example 1.9. A more complex example is the following: If V is a sixdimensional F-vector space and Ω : V × V → F is a non-degenerate alternate bilinear form, then, as a computation shows, the generic rank of M for the linear representation of Sp(Ω) on ∧ 3 V * is 18; see [11] for the details. As dim ∧ 3 V * = 20, it follows that there exist two invariant functions, both of them polynomial functions.
Example 1.10. Given A ∈ gl(2, C) {0}, let X be the infinitesimal generator of the one-parameter group exp(tA), t ∈ C. Let α, β be the eigenvalues of A. We distinguish several cases. If α = β, αβ = 0, then the vector field X admits a first integral in C(x, y) if and only if α −1 β ∈ Q; otherwise, every non-constant first integral of X is a transcendental function. If αβ = 0, then X admits a first integral in C[x, y]. If α = β = 0 and the annihilator polynomial of A is (λ − α) 2 , then X = αx ∂ ∂x + (1 + αy) ∂ ∂y and its basic first integral is the function I = x exp (−αy/x). If the annihilator is λ−α, then X admits a first integral in C(x, y). Finally, If α = β = 0 then the annihilator A is λ 2 and X admits the function x as a first integral.
2 The property (*) studied . Every matrix A ∈ gl(n, F) is identified with the endomorphism on F n to which such matrix corresponds in the basis (v 1 , . . . , v n ). If x = x h v h , then E ij (x) = x j v i , or equivalently E ij (v k ) = δ jk v i , which means that E ij is the matrix with 1 in the entry (i, j) and 0 in the rest of entries. Therefore The Lie algebra of n × n traceless matrices with entries in F is denoted by sl(n, F). The Lie algebra of n × n skew-symmetric matrices with entries in F is denoted by so(n, F). The Lie algebra of 2n × 2n matrices X with entries in F such that X T J n + J n X = 0, where J n = 0 I n −I n 0 , and I n ∈ gl(n, F) is the identity matrix, is denoted by sp(2n, F). By using the formulas (1) and the standard basis for the Lie algebra sl(n, F), i.e., the n 2 − 1 matrices E hi , h = i, h, i = 1, . . . , n; E hh − E 11 , 2 ≤ h ≤ n, we obtain Proposition 2.2. The matrices are a basis for sl(n, F) fulfilling the property (*). Proposition 2.3. If the characteristic of F is either zero or is positive p and p does not divide n, then the identity matrix I ∈ gl(n, F) cannot be written as a sum of square-zero matrices.
Proof.  Proof. If n is odd the result follows from Proposition 2.3. If n = 2m, then are square-zero matrices, and for every 1 ≤ h ≤ n, we have Similarly, by starting with the standard basis for the symplectic Lie algebra sp(2n, F), i.e., we obtain Proposition 2.5. The matrices are a basis for sp(2n, F) fulfilling the property (*).
Similarly, we have Proposition 2.6. The standard basis E hi , 1 ≤ h < i ≤ n, of the Lie subalgebra of strictly upper triangular matrices in gl(n, F) satisfies the property (*).
As for the Lie algebra so(n, F), with basis E hi − E ih , 1 ≤ h < i ≤ n, we have Proposition 2.7. Let x 1 , . . . , x n be the column vectors of a matrix X in so(n, F) of rank r, and let x i 1 , . . . , x ir , 1 ≤ i 1 < . . . < i r ≤ n, be r linearly independent column vectors of X. The necessary and sufficient condition for the square of X to be zero is that the subspace x i 1 , . . . , x ir is totally isotropic with respect to the scalar product ·, · given by v i , v j = δ ij , i, j = 1, . . . , n.
Proof. As X is skew-symmetric, for all i, j = 1, . . . , n, we have Corollary 2.8. If the ground field F is formally real, then the only matrix X in so(n, F) with X 2 = 0 is the zero matrix. 2 and, by virtue of the hypothesis, it follows that the only totally isotropic subspace for ·, · is {0}.
Next we study the property (*) for certain Lie algebras in characteristic 2 following the notations and results of [1]. Assume the characteristic of F is 2, let f : V ×V → F be a bilinear form and let L(f ) ⊆ gl(V ) be its associated Lie subalgebra, i.e.,  In the case [i] the condition (*) does not hold for L(f ) as this condition never holds for gl(m, F).
In the case [ii] the condition (*) does not hold for L(f ) whatever the odd integer n > 1.
In the case of the matrix A in Type 0, L(A) is the Abelian Lie algebra generated by the powers From Proposition 2.6 it follows that the property (*) holds true for the algebra L(A); but the property (*) does not hold for L(B) in Type 0. The Lie algebras L(A) and L(B) in Type 1 do not verify the property (*). As for L(C) ∼ = so(m) in Type 1, the property (*) depends on the nature of the ground field, as we have seen above and the Lie algebra L(D) does verify the property (*) as follows directly from Proposition 2.5.