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)
“I know all those people. I have friendly, social, and criminal relations with the whole lot of them.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“Osteopath—One who argues that all human ills are caused by the pressure of hard bone upon soft tissue. The proof of his theory is to be found in the heads of those who believe it.”
—H.L. (Henry Lewis)