Arnold′s Projective Plane and r-Matrices

1. Introduction

We briefly describe Arnold′s projective geometry [1]. We recall the ordinary projective plane ℙ(ℝ³) over the real field. We replace ℝ³ to the set of the quadratic functions $$ on the canonical symplectic plane (ℝ² : p, q). The quadratic functions form into a Lie subalgebra of the canonical Poisson algebra on the symplectic plane. This Lie algebra is isomorphic to sl(2, ℝ); that is, Arnold introduced a projective plane ℙ(sl(2, ℝ)). By the projection, a (nontrivial) quadratic function $$ corresponds to a point Q on the projective plane, and the Killing form on sl(2, ℝ) corresponds to a quadratic curve on the projective plane, because it is a symmetric bilinear form. Since the Killing form is nondegenerate, the associated curve defines a duality (so-called polar system) between the projective lines and the projective points. Arnold showed that the Poisson bracket $$ corresponds to the pole point of the projective line through Q₁ and Q₂. As an application, it was shown that given a good triangle composed of three points (Q₁, Q₂, Q₃), the three altitudes intersect the same point (altitude theorem). Interestingly, the altitude theorem is shown by the Jacobi identity (see Figure 1).

The monomials of double brackets $$ correspond to the three bold lines, via the line-point duality. By the Jacobi identity, the three monomials are linearly dependent. This implies that the three lines intersect the same point.

We suppose that 2D projective geometry is encoded in the Lie algebra. However some information on the Lie algebra is lost in the process of constructing projective geometry; besides, it is not clear why it has to happen, conceptually. So we will reformulate the Arnold construction by means of lattice theory. Since the lattice is an algebra, the problem becomes more clear. We will prove, when the characteristic of the ground field is not 2, that each 3D simple Lie algebra admits a modular lattice structure. This proposition explains why 2D projective geometry can be encoded in sl(2).

It is crucial to consider the Plücker embedding for the algebraic Arnold construction. When 𝔤 is 3D and simple, the Lie algebra multiplication μ : 𝔤⋀𝔤 → 𝔤, x⋀y ↦ [x, y] is an isomorphism, which induces an isomorphism ℙ(𝔤⋀𝔤)≅ℙ(𝔤). We will show that the line-point duality on ℙ(𝔤) is equivalent with the isomorphism ℙ(𝔤⋀𝔤)≅ℙ(𝔤), up to the Plücker embedding.

As an application we study a projective geometry of triangular r-matrices, when 𝔤 = sl(2). We will prove that the projection of the set of nontrivial triangular r-matrices is equivalent to the pencil of tangent lines on the quadratic curve made from the Killing form. This proposition is equivalently translated as follows: the classical Yang-Baxter equation {r, r} = 0 on sl(2) is equivalent to the quadratic curve on the projective plane.

2. Algebraic Arnold Construction

2.2. Lie Algebra Construction of Projective Plane

Let (𝔤, [−, −]) be a 3D 𝕂-Lie algebra with a nondegenerate symmetric invariant pairing (−, −), where 𝕂 is the ground field char(𝕂) ≠ 2. We assume that [𝔤, 𝔤] = 𝔤, or equivalently, the Lie bracket is an isomorphism from 𝔤⋀𝔤 to 𝔤. This assumption is needed in order to construct projective geometry. Such a Lie algebra is simple. In particular, when 𝕂 is a closed field, 𝔤 is isomorphic to sl(2, 𝕂).

We denote by p(x) the 1D subspace generated by x ∈ 𝔤 and denote by l(x, y) the 2D subspace generated by x, y ∈ 𝔤. We define new cup-and cap-products ⌣^′ and ⌢^′ on Latt(𝔤).

Definition 2.3 (Arnold products). (i) p(x)⌣^′p(y): = p^⊥([x, y]), p(x) ≠ p(y), where ⊥ means the orthogonal space with respect to the invariant pairing (−, −).

(ii) l(x, y)⌢^′l(z, w): = p([[x, y], [z, w]]), l(x, y) ≠ l(z, w).

(iii) S₁⌣^′S₂ : = S₁⌣S₂ and S₁⌢^′S₂ : = S₁⌢S₂, in all other cases.

Proposition 2.4. The set Latt(𝔤) with Arnold products becomes a lattice and it is the same as the classical subspace lattice.

Proof. By the invariancy of the pairing, we have ([x, y], x) = ([x, y], y) = 0. This gives

$$ (2.5)

Hence we have p(x)⌣^′p(y) = p(x)⌣p(y). We prove that l(x, y)∩l(z, w) = p([[x, y], [z, w]]). Since ⊥⊥   = id, we have l^⊥(x, y) = p([x, y]), l^⊥(z, w) = p([z, w]), and p^⊥([[x, y], [z, w]]) = l([x, y], [z, w]). Thus we obtain

$$ (2.6)

which gives l(x, y)⌢^′l(z, w) = l(x, y)⌢l(z, w).

In the following, we omit the “prime” on the Arnold products.

Remark 2.5 (Lie algebra identities versus lattice identities). We put l([x, y]): = l(x, y). Then the Arnold products are coherent with the Lie bracket, namely,

$$ (2.7)

Tomihisa [2] discovered an interesting identity on sl(2, ℝ):

$$ (2.8)

where y, z are fixed. The Tomihisa identity induces a lattice identity

$$ (2.9)

where l_i : = l(x_i), l_y : = l(y), and l_z : = l(z). In a study by Aicardi in [3], it was shown that the projection of Tomihisa identity is equivalent to the Pappus theorem. (The author also proved this proposition around winter 2007.) We leave it to the reader to write down the lattice identity associated with the Jacobi identity.

Given a coordinate (ξ₁, ξ₂, ξ₃) on the projective plane, the symmetric pairing is regarded as a defining equation of a nondegenerate quadratic curve (so-called polar system):

$$ (2.10)

We give an example of the quadratic curve made from the pairing.

Example 2.6 (see, [1, 2]). We assume that 𝔤 : = sl(2, ℝ) generated by the standard basis (X, Y, H) satisfying the following relations:

$$ (2.11)

The symmetric pairing is equivalent with the Killing form. We define the scale of the form by

$$ (2.12)

and all others zero. We set a point Q : = ξ₁Y + ξ₂H + ξ₃X, ξ₁, ξ₂, ξ₃ ∈ ℝ. It is on the quadratic curve, that is, (Q, Q) = 0 if and only if ξ₁ξ₃ − (ξ₂) ² = 0, and this condition is equivalent with (x₀) ² = (x₁) ² + (x₂) ², via the coordinate transformation

$$ (2.13)

Hence the quadratic curve is regarded as a circle on the projective plane with coordinate [1 : x₁ : x₂].

Remark 2.7. The pairing induces a metric (−++) on (ℝ³ : x₀, x₁, x₂). It is well known that the inside of the circle (so-called timelike subspace) is a hyperbolic plane. The altitude theorem is strictly a theorem on the hyperbolic plane because a metric is needed to define the notion of altitude.

Given a point p = (p₁, p₂, p₃), a line is defined by

$$ (2.14)

This line is called the polar line of the point; conversely the point is called the pole of the line. Namely, the orthogonal space p : = l^⊥ is the pole of the line l.

Corollary 2.8 (see [1].)Given a line l(x, y), p([x, y]) is the pole of the line.

Figure 2 is depicting the duality defined by an ellipse.

The line l is the polar line of the point p which is inside an ellipse. The line and point are connected by the tangent lines and chords of the ellipse. This figure has been drawn in a book by Kawada in [4].

We did not use Jacobi identity in this section. So we will discuss the Jacobi identity on the Lie algebra in the next section.

2.4. Duality Principle

Let μ be the Lie algebra structure on 𝔤; that is, μ : 𝔤⋀𝔤 → 𝔤, x⋀y ↦ [x, y]. Since μ is an isomorphism, it induces a lattice isomorphism

$$ (2.20)

Let Line : = {p(x)⌣p(y)} be the set of all lines in Latt(𝔤). One can define an injection pl : Line → Latt(𝔤⋀𝔤) as

$$ (2.21)

This mapping is called a Plücker embedding.

Proposition 2.10. The diagram below is commutative:

(2.22)

where Point is the set of points on the projective plane and ⊥ is the duality correspondence.

Namely, the Lie algebra multiplication is equivalent to the duality principle.

3. r-Matrices

Endnotes