Meaning of 'Triple'

**gsrith** · June 19th 2009, 03:47 PM

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 $A=(a_{ij})_{i.j=1}^n$ of rank $l$ ; and define a realization of A to be the triple $(\mathfrak{h},\Pi,\Pi^v)$ where $\mathfrak{h}$ is a complex vector space of finite dimension, $\Pi=\{\alpha_1,\alpha_2...\alpha_n\}$ and $\Pi^v=\{\alpha_1^v,\alpha_2^v...\alpha_n^v\}$ are indexed subsets of $\mathfrak{h}*$ and $\mathfrak{h}$ respectively such that they satisfy following properties
1) Both sets are linearly independent
2) $\langle\alpha_i^v,\alpha_j\rangle=a_{ij}$
where $\langle,\rangle :\mathfrak{h}\times\mathfrak{h}*\rightarrow C$ denotes the pairing $\langle h, \alpha \rangle=\alpha(h)$

"

**NonCommAlg** · June 19th 2009, 07:41 PM

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 $A=(a_{ij})_{i.j=1}^n$ of rank $l$ ; and define a realization of A to be the triple $(\mathfrak{h},\Pi,\Pi^v)$ where $\mathfrak{h}$ is a complex vector space of finite dimension, $\Pi=\{\alpha_1,\alpha_2...\alpha_n\}$ and $\Pi^v=\{\alpha_1^v,\alpha_2^v...\alpha_n^v\}$ are indexed subsets of $\mathfrak{h}*$ and $\mathfrak{h}$ respectively such that they satisfy following properties
1) Both sets are linearly independent
2) $\langle\alpha_i^v,\alpha_j\rangle=a_{ij}$
where $\langle,\rangle :\mathfrak{h}\times\mathfrak{h}*\rightarrow C$ denotes the pairing $\langle h, \alpha \rangle=\alpha(h)$

"

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 $V$ is a finite dimensional vector space over $\mathbb{C}$ and $A$ an $n \times n$ matrix

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

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

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

**gsrith** · June 19th 2009, 08:02 PM

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 $C$ is a triple $(P,L,I)$ where $P$ is a set of points, $L$ is a set of lines and $I$ is a subset of $P\times L$ .

**HallsofIvy** · June 20th 2009, 04:51 AM

A "triple" is simply "three things". It applies here because $(\mathfrak{h},\Pi,\Pi^v)$ consists of three things. It is not necessary to say, here, "ordered triple", as in $R^3$ , 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!

)

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