Originally Posted by

**gsrith** I found this description in the book on Kac Moody Algebra by Wan. Can someone explain whats meant by triple?

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Title: Realization of a Complex nXn Matrix A

We start with an arbitrary complex n X n matrix $\displaystyle A=(a_{ij})_{i.j=1}^n$ of rank $\displaystyle l$; and define a realization of A to be the triple $\displaystyle (\mathfrak{h},\Pi,\Pi^v)$ where $\displaystyle \mathfrak{h}$ is a complex vector space of finite dimension, $\displaystyle \Pi=\{\alpha_1,\alpha_2...\alpha_n\}$ and $\displaystyle \Pi^v=\{\alpha_1^v,\alpha_2^v...\alpha_n^v\}$ are indexed subsets of $\displaystyle \mathfrak{h}*$ and $\displaystyle \mathfrak{h}$ respectively such that they satisfy following properties

1) Both sets are linearly independent

2) $\displaystyle \langle\alpha_i^v,\alpha_j\rangle=a_{ij}$

where $\displaystyle \langle,\rangle :\mathfrak{h}\times\mathfrak{h}*\rightarrow C $ denotes the pairing $\displaystyle \langle h, \alpha \rangle=\alpha(h)$

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