Relations With Representation Theory
Consider the following slightly more general context. Suppose that n and m are two integers and xij for i = 1,...,n,j = 1,...,m, be commuting variables. Redefine Eij by almost the same formula:
with the only difference that summation index a ranges from 1 to m. One can easily see that such operators satisfy the commutation relations:
Here denotes the commutator . These are the same commutation relations which are satisfied by the matrices which have zeros everywhere except the position (i,j), where 1 stands. ( are sometimes called matrix units). Hence we conclude that the correspondence defines a representation of the Lie algebra in the vector space of polynomials of xij.
Read more about this topic: Capelli's Identity
Famous quotes containing the words relations with, relations and/or theory:
“Society does not consist of individuals but expresses the sum of interrelations, the relations within which these individuals stand.”
—Karl Marx (1818–1883)
“If one could be friendly with women, what a pleasure—the relationship so secret and private compared with relations with men. Why not write about it truthfully?”
—Virginia Woolf (1882–1941)
“There is in him, hidden deep-down, a great instinctive artist, and hence the makings of an aristocrat. In his muddled way, held back by the manacles of his race and time, and his steps made uncertain by a guiding theory which too often eludes his own comprehension, he yet manages to produce works of unquestionable beauty and authority, and to interpret life in a manner that is poignant and illuminating.”
—H.L. (Henry Lewis)