We will explain Arnold's 2-dimensional (shortly, 2D) projective geometry (Arnold, 2005) by means of lattice theory. It will be shown that the projection of the set of nontrivial triangular
We briefly describe Arnold's projective geometry [
The monomials of double brackets
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
It is crucial to consider the
As an application we study a projective geometry of triangular
A lattice is, by definition, a set equipped with two commutative associative multiplications,
Let
The subspace lattices are
When
The 1D subspaces are regarded as projective points, 2D subspaces are projective lines and so on. The zero space
Let
We denote by
(i)
(ii)
(iii)
The set
By the invariancy of the pairing, we have
In the following, we omit the “prime” on the Arnold products.
We put
Given a coordinate
We assume that
The pairing induces a metric
Given a point
Given a line
Figure
The line
We did not use Jacobi identity in this section. So we will discuss the Jacobi identity on the Lie algebra in the next section.
Let
When By the invariancy of the pairing, we have
Assume that
The proposition above indicates that the Jacobi identity is a priori invested in the projective plane.
Let
This mapping is called a
The diagram below is commutative:
Namely, the Lie algebra multiplication is equivalent to the duality principle.
In this section, we assume that
When
The set of points
Since The set of lines
If
A line is tangent to the quadratic curve if and only if the pole is on the line, and the tangent point is the pole of the line (see Figure
The author would like to express his thanks to Professor Akira Yoshioka and Dr. Toshio Tomihisa for their helpful comments.
The complement is not unique in general.