An important case of vector-valued differential forms are Lie algebra-valued forms. These are -valued forms where is a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.
Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie algebra-valued forms can be composed with the bracket operation to obtain another Lie algebra-valued form. This operation is denoted by to indicate both operations involved, or often just . For example, if and are Lie algebra-valued one forms, then one has
With this operation the set of all Lie algebra-valued forms on a manifold M becomes a graded Lie superalgebra.
The operation can also be defined as the bilinear operation on satisfying by the formula
for all and .
The alternative notation, which resembles a commutator, is justified by the fact that if the Lie algebra is a matrix algebra then is nothing but the graded commutator of and, i. e. if and then
where are wedge products formed using the matrix multiplication on .
Read more about this topic: Vector-valued Differential Form
Famous quotes containing the words lie and/or forms:
“Even as fog continues to lie in the valleys, so does ancient sin cling to the low places, the depressions in the world consciousness.”
—Dewitt Bodeen (1908–1988)
“It would be easy ... to regard the whole of world 3 as timeless, as Plato suggested of his world of Forms or Ideas.... I propose a different view—one which, I have found, is surprisingly fruitful. I regard world 3 as being essentially the product of the human mind.... More precisely, I regard the world 3 of problems, theories, and critical arguments as one of the results of the evolution of human language, and as acting back on this evolution.”
—Karl Popper (1902–1994)