# Thread: Meaning of 'Triple'

1. ## Meaning of 'Triple'

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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2. Originally Posted by gsrith
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 $\displaystyle A=(a_{ij})_{i.j=1}^n$ 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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that's a bad notation! i'm not sure i understand your question. so i'll just put the definition in a simpler language: suppose $\displaystyle V$ is a finite dimensional vector space over $\displaystyle \mathbb{C}$ and $\displaystyle A$ an $\displaystyle n \times n$ matrix

with entries from $\displaystyle \mathbb{C}.$ a realization of $\displaystyle A$ is a representation of $\displaystyle A$ as $\displaystyle A=[\alpha_j(v_i)],$ where the sets $\displaystyle \{v_1, \cdots , v_n \} \subset V$ and $\displaystyle \{\alpha_1, \cdots , \alpha_n \} \subset V^*$ are linearly independent.

this realization is completely determinded by three factors here: the vector space $\displaystyle V$ and the sets $\displaystyle \mathcal{B}=\{v_1, \cdots , v_n \}$ and $\displaystyle \mathcal{C}=\{\alpha_1, \cdots , \alpha_n \}.$ so we may also say that a realization of $\displaystyle A$ is the triple

$\displaystyle (V, \mathcal{B}, \mathcal{C}).$

3. My question was actually about what triple meant. Wikipedia gives a category theory related definition, which I could not understand(don't even know whether its talking about the same thing). I have encountered the usage elsewhere too.
For example in the definition of an 'incidence structure':
An incidence structure $\displaystyle C$ is a triple $\displaystyle (P,L,I)$ where $\displaystyle P$ is a set of points, $\displaystyle L$ is a set of lines and $\displaystyle I$ is a subset of $\displaystyle P\times L$.

4. A "triple" is simply "three things". It applies here because $\displaystyle (\mathfrak{h},\Pi,\Pi^v)$ consists of three things. It is not necessary to say, here, "ordered triple", as in $\displaystyle R^3$, because there are three different kinds of things that cannot be confused.

(I keep forgetting not everyone on this forum is as fluent in English as I is!)