Lie algebra extension
A large portion towards the end is devoted to background material for applications of Lie algebra extensions, both in mathematics and in physics, in areas where they are actually useful. A parenthetical link, (background material), is provided where it might be beneficial.
Lie algebras that are ingredients in an extension will, without comment, be taken to be over the same field.
The summation convention applies, including sometimes when the indices involved are both upstairs or both downstairs.
An ideal is a subalgebra, but a subalgebra is not necessarily an ideal. A trivial extension is thus a split extension.
As with the trivial extension, this property generalizes to the definition of a split extension.
The "negative" result of the preceding theorem indicates that one must, at least for semisimple Lie algebras, go to infinite-dimensional Lie algebras to find useful applications of central extensions. There are indeed such. Here will be presented affine Kac–Moody algebras and Virasoro algebras. These are extensions of polynomial loop-algebras and the Witt algebra respectively.
But these forms the basis of a vector space in which every form has the right properties.
For basis elements, suitably normalized and with antisymmetric structure constants, one has
In addition, assume the basis on the underlying finite-dimensional simple Lie algebra has been chosen so that the structure coefficients are antisymmetric in all indices and that the basis is appropriately normalized. Then one immediately through the definitions verifies the following commutation relations.
These are precisely the short-hand description of an untwisted affine Kac–Moody algebra. To recapitulate, begin with a finite-dimensional simple Lie algebra. Define a space of formal Laurent polynomials with coefficients in the finite-dimensional simple Lie algebra. With the support of a symmetric non-degenerate alternating bilinear form and a derivation, a 2-cocycle is defined, subsequently used in the standard prescription for a central extension by a 2-cocycle. Extend the derivation to this new space, use the standard prescription for a split extension by a derivation and an untwisted affine Kac–Moody algebra obtains.
It is desirable to get rid of the central charge on the right hand side. To do this define
With β = 0 it is possible to change basis (or modify the 2-cocycle by a 2-coboundary) so that
Because of the continuum of operators, and because of the delta functions, it is desirable to express these relations instead in terms of the quantized versions of the Virasoro modes, the Virasoro operators. These are calculated to satisfy
They are interpreted as creation and annihilation operators acting on Hilbert space, increasing or decreasing the quantum of their respective modes. If the index is negative, the operator is a creation operator, otherwise it is an annihilation operator. (If it is zero, it is proportional to the total momentum operator.) In view of the fact that the light cone plus and minus modes were expressed in terms of the transverse Virasoro modes, one must consider the commutation relations between the Virasoro operators. These were classically defined (then modes) as
In order to adequately discuss extensions, structure that goes beyond the defining properties of a Lie algebra is needed. Rudimentary facts about these are collected here for quick reference.
For the present purposes, consideration of a limited portion of the theory Lie algebra cohomology suffices. The definitions are not the most general possible, or even the most common ones, but the objects they refer to are authentic instances of more the general definitions.
and having a property resembling the Jacobi identity called the Jacobi identity for 2-cycles,
These notions generalize in several directions. For this, see the main articles.
for the basis elements, where the summation symbol has been rationalized away, the summation convention applies. The placement of the indices in the structure constants (up or down) is immaterial. The following theorem is useful:
This is resembling Riesz representation theorem and the proof is virtually the same. The Killing form has the property
These rather formidable equations simplify considerably with a clever choice of parametrization called the light cone gauge. In this gauge, the equations of motion become
the ordinary wave equation. The price to be paid is that the light cone gauge imposes constraints,
so that one cannot simply take arbitrary solutions of the wave equation to represent the strings. The strings considered here are open strings, i.e. they don't close up on themselves. This means that the Neumann boundary conditions have to be imposed on the endpoints. With this, the general solution of the wave equation (excluding constraints) is given by