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A Hyperhermitian condition

  1. May 7, 2016 #1
    Let ##V## be a quaternionic vector space with quaternionic structure ##\{I,J,K\}##. One can define a Riemannian metric ##G## and hyperkahler structure ##\{\Omega^{I},\Omega^{J}, \Omega^{K}\}##. Do this inner product
    $$\langle p,q \rangle := G(p,q)+i\Omega^{I}(p,q)+j\Omega^{J}(p,q)+k\Omega^{K}(p,q)$$
    really satisfy hyperhermitian condition?
     
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  3. May 9, 2016 #2

    chiro

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    Hey Leditto.

    Could you please (for those of us like myself unfamiliar with the field and terminology) give a description of the condition?

    Also - what is a quaternionic structure? I know what a quaternion is - is it just a tensor product of three quaternions?
     
  4. May 9, 2016 #3

    Ben Niehoff

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    On a vector space, a quaternionic structure is a set of three linear operators ##I,J,K## such that

    $$I^2 = J^2 = K^2 = IJK = -\mathrm{id}, \quad IJ = K, \quad JK = I, \quad KI = J.$$
    However, I've not heard the word "hyperhermitian" before.
     
  5. May 9, 2016 #4

    jim mcnamara

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  6. May 10, 2016 #5

    Thanks for your response, Niehoff.

    In complex case, Hermitian condition is described by $$\langle I u,I v \rangle=\langle u,v \rangle.$$ Quaternionic analogue of that condition is called hyperhermitian condition and defined by $$\langle I u,I v \rangle=\langle J u,J v \rangle=\langle K u,K v \rangle = \langle u,v \rangle.$$ In addition, there are metric compatibilities condition that make vector space ##V## a hyperkahler manifold, $$G(Iu,v)=\Omega^{I}(u,v),\quad G(Ju,v)=\Omega^{J}(u,v),\quad G(Ku,v)=\Omega^{K}(u,v).$$ I've checked that hyperhermitian condition can't be fulfilled by defining $$\langle u,v \rangle=G(u,v)+i\,\Omega^{I}(u,v)+j\,\Omega^{J}(u,v)+k\,\Omega^{K}(u,v).$$ My calculation:
    \begin{eqnarray*}
    \langle I u,I v \rangle&=&G(Iu,Iv)+i\,\Omega^{I}(Iu,Iv)+j\,\Omega^{J}(Iu,Iv)+k\,\Omega^{K}(Iu,Iv)\\
    &=&\Omega^{I}(u,Iv)+i\,G(I^2u,Iv)+j\,G(JIu,Iv)+k\,G(KIu,Iv)\\
    &=&-\Omega^{I}(Iv,u)-i\,G(Iv,u)-j\,G(Ku,Iv)+k\,G(Ju,Iv)\\
    &=&-G(I^2v,u)-i\,\Omega^{I}(v,u)-j\,\Omega^{K}(u,Iv)+k\,\Omega^{J}(u,Iv)\\
    &=&G(u,v)+i\,\Omega^{I}(u,v)+j\,\Omega^{K}(Iv,u)-k\,\Omega^{J}(Iu,v)\\
    &=&G(u,v)+i\,\Omega^{I}(u,v)+j\,G(KIv,u)-k\,G(JIv,u)\\
    &=&G(u,v)+i\,\Omega^{I}(u,v)+j\,G(Jv,u)+k\,G(Kv,u)\\
    &=&G(u,v)+i\,\Omega^{I}(u,v)-j\,\Omega^{J}(u,v)-k\,\Omega^{K}(u,v)\\
    &\neq& \langle u, v \rangle
    \end{eqnarray*}

    Did I make something wrong in my elaboration? Can You spot it?
     
    Last edited: May 10, 2016
  7. May 10, 2016 #6
    Thanks Jim McNamara

    I focus only on a quaternionic vector space case which can be seen as a (linear) hyperkahler manifold.
     
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