Rational Representations of Some Algebraic Varieties of M-Orthogonal Groups

Mircea Geanău

Abstract

The purpose of this paper is the study of the algebraic variety of the Morthogonal group, insisting on the rational representations of M-orthogonal matrices. We shall follow the study of a certain subgroup of the M-orthogonal group and we will give a parametric rational representation of this variety.

Mathematics Subject Classification: 20G20 Key words: algebraic variety, rational representation, M-orthogonal group.

Let GL(n, R) be the group of real non-singular matrices of n-th order and let M be a fixed matrix of n-th order that doesn’t necessarily belong to GL(n, R). We shall consider the following nonempty set denoted by

G_{M} = {A ∈ GL(n, R)|AMÃ = M},

where Ã is the transpose of A. It is almost obvious that the set G_{M} having as a law composition the multiplication of the matrices is a group.

By definition, G_{M} is the M-orthogonal group. In the study of the group G_{M}, two cases, M non-singular and M singular assert themselves.

a) Let suppose that M is non-singular and symmetric.

Because it exists 1 + 2 + ... + n = \frac{n(n + 1)}{2} independent relations, it results that

the variety depends of n^{2} − \frac{n(n + 1)}{2} = \frac{n(n − 1)}{2} parameters.

Proposition 1. If M is non-singular and symmetric matrix, the relation A = (M + X)^{−1}(M − X) with X̃ = −X and M^{2}X = XM^{2}, defines a rational parametric representation of the M-orthogonal group G_{M} in the neighbourhood determined by the condition |M + X| ≠ 0. Proof. We must show that the equality

(1)  (M + X)^{−1}(M − X)M(M + X)(M − X)^{−1}

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is true.

Because M^{2}X = XM^{2}, the equality

(M − X)M(M + X) = (M + X)M(M − X)

is fulfilled which implies the valability of the relation (1).

Note: By keeping fulfilled the conditions from Proposition 1, it shows as above that the relation A = (M + X)(M − X)^{−1}, with |M − X| ≠ 0, gives a parametric rational representation of the M-orthogonal group in the neighbourhood determined by |M − X| ≠ 0.

From the equality A = (M + X)^{−1}(M − X), multiplied at left by (M + X) it results that

X = M(E − A)(E + A)^{−1} with |A + E| ≠ 0

and E is the identity matrix. Example 1. Taking

M = E = ( \begin{matrix}1 & 0 & … & 0 \\ 0 & 1 & … & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & … & 1 \end{matrix} )

it is obtained a parametric representation of the orthogonal group,

A = (E + X)^{−1}(E − X) with |E + A| ≠ 0, A ∈ O(n, R).

2. In the particular case

M = ( \begin{matrix}−1 & & & & & 0 \\ & −1 \\ & & −1 \\ & & & 1 \\ & & & & 1 \\ 0 & & & & & 1 \end{matrix} ) ,

where -1 appears p times, with M^{2} = E, it is obtained the parametric representation of the pseudo-orthogonal group.

b) Let’s take the case when the matrix M is non-singular and antisymmetric. Because M = AMÃ is antisymmetric it results that the dimension n = 2p and

it exists \frac{n(n − 1)}{2} independent relations and in that case the algebraic variety has

n^{2} − \frac{n(n − 1)}{2} = \frac{n(n + 1)}{2} parameters. Considering X a symmetric matrix and making similar reasonings with those from above its is proved just like in the case of Proposition 1 the following result Proposition 2. If the matrix M is non-singular and antisymmetric, the equality A = (M + X)^{−1}(M − X), with X̃ = X and M^{2}X = XM^{2}, defines a rational parametric representation of the M-orthogonal group G_{M} in the neighbourhood limited by |M + X| ≠ 0. Example. Let

M = ( \begin{matrix}0 & E \\ … & … \\ −E & 0 \end{matrix} )

be an antisymmetric matrix of 2n-th order where E is the identity matrix of nth order. Because are fulfilled the conditions of Proposition 2 we get a parametric representation of the symplectic group S_{p}(2n, R).

c) Let’s suppose that M is a singular and symmetric matrix. There are

1 + 2 + ... + n = \frac{n(n + 1)}{2}

independent and distinct relations so the variety has

n^{2} − \frac{n(n + 1)}{2} = \frac{n(n − 1)}{2}

parameters.

In the case when n = 2, due to the fact that M^{2} is also symmetric and singular, if it exists an antisymmetric matrix X with M^{2}X = XM^{2}, it results that M^{2} = 0, this one implies M = 0, which shows that it doesn’t exist a parametric representation of the form A = (M + X)^{−1}(M − X).

If n ≥ 3, using a similar computation with the one used at Proposition 1, it is shown that the equality A = (M + X)^{−1}(M − X), with X̃ = −X and M^{2}X = XM^{2}, defines a rational parametric representation of the M- orthogonal group G_{M} in the neighbourhood given by the |M + X| ≠ 0.

In the hypothesis that M is singular and antisymmetric, n ≥ 3, analogous to the Proposition 2, the relation A = (M +X)^{−1}(M −X), with X̃ = X and M^{2}X = XM^{2}, defines a rational parametric representation of the group G_{M} in the neighbourhood defined by |M + X| ≠ 0.

Keeping the above results and notations we shall treat a more general case, that of the subset of G_{M}, noted H_{M,N} defined such that,

H_{M,N} = {A ∈ GL(n, R), AMÃ = M, AN = NA},

where M and N are two fixed matrices of n-th order.

It is obvious that H is nonempty and is a subgroup of G_{M}. We shall try to find rational parametric representations of these varieties. Because M and N are two fixed matrices, without specific forms, we shall assume that the variety obtained has p parameters.

Analogous to the M-orthogonal group, we shall treat more situations concerning the fact that M is non-singular or singular and MN = NM or MN = −NM. Proposition 3. Let H_{M,N} be the subgroup of the M-orthogonal group defined by

\{ \begin{matrix}AMÃ = M \\ AN = NA. \end{matrix}

If M is non-singular and symmetric and N is involutive (N = N^{−1}) and verifies the equality MN = NM, then the algebraic variety V_{p} is rational, the formula A = (M + X)^{−1}(M − X), with X̃ = −X and XN = NX, XM^{2} = M^{2}X, |M + X| ≠ 0, defines a rational parametric representation of this variety in the neighbourhood determined by the condition |M + X| ≠ 0. Proof. From Proposition 1, we know that it is verified the equality AMÃ = M. We shall show that there exists the relation AN = NA if are fulfilled the conditions from Proposition 3.

The relation AN = NA is equivalent to (M +X)^{−1}(M −X)N = N(M +X)^{−1}(M − X). Because MN = NM and XN = NX, it results (M − X)N = N(M − X), (M + X)N = N(M + X). Then

(M + X)^{−1}(M − X)N = (M + X)^{−1}N(M − X) =
= (M + X)^{−1}N^{−1}(M − X) = [N(M + X)^{−1}](M − X) =
= [(M + X)N]^{−1}(M − X) = N^{−1}(M + X)^{−1}(M − X) =
= N(M + X)^{−1}(M − X)

which shows that AN = NA. In the hypothesis MN = −NM, we have the following results Proposition 4. Let H_{M,N} be the subgroup of the M-orthogonal group G_{M}, defined by the relations AMÃ = M, AN = NA.

If M is symmetric non-singular matrix and N = N^{−1} verifies the equality MN = −NM, then the equality A = (M + X)^{−1}(M − X), with X̃ = −X and

\{ \begin{matrix}XM^{2} = M^{2}X \\ XN = −NX \end{matrix}

defines a rational parametric representation of the group H_{M,N} in the neighbourhood determined by |M + X| ≠ 0. Proof. Analogous to the Proposition 3, we must show that is fulfilled the equality AN = NA. This one shows that the equality (M + X)^{−1}(M − X)N = N(M + X)^{−1}(M − X) is true.

From those conditions it results that the relation N(M − X) = −(M − X) is true. The following equalities are true:

(M + X)^{−1}(M − X)N = −(M + X)^{−1}N(M − X) =
−(M + X)^{−1}N^{−1}(M − X) = −[N(M + X)^{−1}](M − X) =
= [(M + X)N]^{−1}(M − X) = N^{−1}(M + X)^{−1}(M − X) =
= N(M + X)^{−1}(M − X).

We’ll continue to study the case when M is non-singular and antisymmetric. Proposition 5. If M is a non-singular antisymmetric matrix and N is an involutive matrix with MN = −NM, then the formula

A = (M + X)^{−1}(M − X) with |M + X| ≠ 0 and XM^{2} = M^{2}X,

XN = −NX, X̃ = X, defines a rational parametric representation of the subgroup H_{M,N}. Proof. We know that the equality AMÃ = M is verified (Proposition 2).

It is remained to be shown that the equality

(2)  (M + X)^{−1}(M − X)N = N(M + X)^{−1}(M − X)

is true. We shall show that the right member of the equality (2) is the same as that from the left. We have then

N(M + X)^{−1}(M − X) = N^{−1}(M + X)^{−1}(M − X) =
= [(M + X)N]^{−1}(M − X) = (−NM − NX)^{−1}(M − X) =
= [−N(M + X)]^{−1}(M − X) = −(M + X)^{−1}N(M − X) =
= −(M + X)^{−1}(−MN − NX) = (M + X)^{−1}(MN + NX) =
= (M + X)^{−1}(MN − XN) = (M + X)^{−1}(M − X)N

which shows that the equality (2) it’s true.

Example. Let

M_{0} = ( \begin{matrix}I_{2} & 0 & … & 0 \\ 0 & I_{2} & … & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & … & I_{2} \end{matrix} ) and N_{0} = ( \begin{matrix}J_{2} & 0 & … & 0 \\ 0 & J_{2} & … & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & … & J_{2} \end{matrix} )

be cellular matrices of 2n-th order with

I_{2} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , J_{2} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

It verifies that M_{0}N_{0} = −N_{0}M_{0}, N_{0}^{2} = E_{2n}, where E_{2n} is the identity matrix of

2n-th order. In this particular case the subgroup H_{M_{0},N_{0}} depends of n^{2} parameters. If X is a non-singular symmetric matrix of 2n-th order which verifies the relation XN_{0} = −N_{0}X, then the variety of the group H_{M_{0},N_{0}} formed by the matrices A which AMÃ = M_{0}, AN_{0} = N_{0}A, admits a rational parametric representation of the form A = (M_{0} + X)^{−1}(M_{0} − X) in the neighbourhood determined by |M_{0} + X| ≠ 0. Remark. Let H_{M,N} be the subgroup of G_{M} and let B ∈ G_{M} be. In these condition the set S = {B^{−1}AB|A ∈ H_{M,N}} is a subgroup of G_{M}, isomorphic which H_{M,N}.

Obviously the set S is nonempty. Let B^{−1}A_{1}B ∈ S and B^{−1}A_{2}B ∈ S, then

(B^{−1}A_{1}B)(B^{−1}A_{2}B)^{−1} = B^{−1}A_{1}BB^{−1}A_{2}^{−1}B = B^{−1}(A_{1}A_{2})^{−1}B ∈ S

because H_{M,N} is a subgroup of G_{M}. The function f : H_{M,N} → S defined by f(A) =

B^{−1}AB is an isomorphism of H_{M,N} on S.

74 M.Geanău

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University Politehnica of Bucharest Department of Mathematics I Splaiul Independenţei 313 77206 Bucharest, Romania