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Math Help - Meaning of 'Triple'

  1. #1
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    Post Meaning of 'Triple'

    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)

    "
    Last edited by gsrith; June 19th 2009 at 04:56 PM. Reason: missed a word!!
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  2. #2
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    Quote Originally Posted by gsrith View Post
    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}).
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  3. #3
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    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.
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  4. #4
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    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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