What is the geometrical meaning of the central extension of the algebra
of diffeomorphisms of the circle?
The geometrical meaning of the central extension of the algebra of diffeomorphisms of the circle can be explained as follows:
To understand this, we first need to understand what the algebra of diffeomorphisms of the circle is. The diffeomorphisms of the circle refer to smooth, invertible mappings from the circle to itself. These mappings can be composed to form a group, known as the diffeomorphism group.
The algebra of diffeomorphisms of the circle is the Lie algebra associated with this group. It consists of vector fields on the circle, which can be thought of as infinitesimal generators of the diffeomorphisms.
Now, a central extension of a Lie algebra is a way to extend the Lie algebra by adding a new element that commutes with all other elements. In other words, the extended Lie algebra has an additional generator that does not affect the Lie bracket operation.
In the context of the diffeomorphisms of the circle, a central extension of the algebra can be interpreted geometrically as the addition of an extra degree of freedom that is preserved by all diffeomorphisms.
This extra degree of freedom is often associated with the concept of winding numbers. The winding number is a topological invariant that measures how many times a curve winds around the circle. It can be thought of as a "counting" of the rotations or twists of a curve around the circle.
The central extension of the algebra of diffeomorphisms introduces a new generator, called the central element, which represents the winding number. This central element commutes with all other elements in the algebra, meaning that it does not affect the Lie bracket operation.
So, geometrically, the central extension of the algebra of diffeomorphisms of the circle can be understood as the incorporation of winding numbers as an additional symmetry, allowing for a more complete description of the transformations of the circle.