I found this description in the book on Kac Moody Algebra by Wan. Can someone explain whats meant by triple?
Title: Realization of a Complex nXn Matrix A
We start with an arbitrary complex n X n matrix of rank ; and define a realization of A to be the triple where is a complex vector space of finite dimension, and are indexed subsets of and respectively such that they satisfy following properties
1) Both sets are linearly independent
where denotes the pairing