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:
“In the middle of the night, as indeed each time that we lay on the shore of a lake, we heard the voice of the loon, loud and distinct, from far over the lake. It is a very wild sound, quite in keeping with the place and the circumstances of the traveler, and very unlike the voice of a bird. I could lie awake for hours listening to it, it is so thrilling.”
—Henry David Thoreau (1817–1862)
“That food has always been, and will continue to be, the basis for one of our greater snobbisms does not explain the fact that the attitude toward the food choice of others is becoming more and more heatedly exclusive until it may well turn into one of those forms of bigotry against which gallant little committees are constantly planning campaigns in the cause of justice and decency.”
—Cornelia Otis Skinner (1901–1979)