I have been reading some notes: Vector Symmetries by Lyubashenko and I'm looking to understand some of the works to apply to my own research.

In it, he writes a matrix operation in a basis independent way so that it may be used in the context of operators. Essentially, he defines the "sharp" operation on a matrix $M: \mathcal{V} \otimes \mathcal{V} \to \mathcal{V} \otimes \mathcal{V}$ (where $\mathcal{V}$ is a finite dimensional vector space) as $$(M^{\#})^{\alpha \beta}_{\delta \gamma} = M^{\beta \gamma}_{\alpha \delta}$$

He then goes on to describe the same operation on an operator $T: \mathcal{H} \otimes \mathcal{H} \to \mathcal{H} \otimes \mathcal{H}$ ($\mathcal{H}$ a Hilbert space). He says "Let $T^{\#} : \mathcal{H}\otimes \mathcal{H}^* \to \mathcal{H}^*\otimes \mathcal{H}$ ($\mathcal{H}^*$ the duel space of $\mathcal{H}$) and let an element $h_1^* \otimes h_2^* \otimes h_1 \otimes h_2 \in \mathcal{H}^* \otimes \mathcal{H}^* \otimes \mathcal{H}\otimes \mathcal{H}$ represent $T$. After a cyclic rearrangement of factors, it yields $h_2 \otimes h_1^* \otimes h_2^* \otimes h_1$ that represents the operator $T^{\#}$ via $$T^{\#}(\eta \otimes \xi^*) = \langle h_2, \xi^*\rangle \langle \eta, h_1^*\rangle h_1^*\otimes h_2.$$ "

My questions are:

  • What does it mean for an element to "represent an operator"?
  • Does this "cyclic rearrangement of factors" corresponds to the 'rotation' of indices in the matrix case?
  • How does this formula definition get applied in calculation? For example, we have the operator $F$ defined as $F(\eta \otimes \xi) = \xi \otimes \eta$. This operator should be invariant under the sharp operation, i.e $F^{\#} = F$, but how do we show this using the expression above?
  • Furthermore, how is this definition consistent with the matrix case?

I apologise for the list of questions, and I'm very grateful for help.

  • $\begingroup$ I know Volodymyr Lyubashenko. You can reach him via this data. $\endgroup$ – Alex Ravsky Apr 13 at 16:42

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