# Tensor product over 3 (or more) vector spaces

## Homework Statement

I have the operators

$D_{\beta}:V_{\beta}\rightarrow V_{\beta}$
$R_{\beta\alpha 1}: V_{\beta} \otimes V_{\alpha 1} \rightarrow V_{\beta}\otimes V_{\alpha 1}$
$R_{\beta\alpha 2}: V_{\beta} \otimes V_{\alpha 2} \rightarrow V_{\beta}\otimes V_{\alpha 2}$

where each of the vector spaces are copies of $\mathbb{C}^2$

## The Attempt at a Solution

I want to write the product $D_{\beta}R_{\beta\alpha 1}R_{\beta\alpha 2}$, which makes sense on the space $V_{\beta} \otimes V_{\alpha 1} \otimes V_{\alpha 2}$.

So I order to act on the full space I write $D$ as $D_{\beta}\otimes I_{\alpha 1} \otimes I_{\alpha 2}$
and write $R_{\beta\alpha 1}$ as $R_{\beta\alpha 1} \otimes I_{\alpha 2}$, where $I_{\alpha 1}$ and $I_{\alpha 2}$ are the identity operators in $V_{\alpha 1}$ and $V_{\alpha 2}$.

My problem is that I don't know how to write $R_{\beta\alpha 2}$ since I'd basically have to stick and identity "in the middle".

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Stephen Tashi

## Homework Statement

I have the operators

$D_{\beta}:V_{\beta}\rightarrow V_{\beta}$
$R_{\beta\alpha 1}: V_{\beta} \otimes V_{\alpha 1} \rightarrow V_{\beta}\otimes V_{\alpha 1}$
$R_{\beta\alpha 2}: V_{\beta} \otimes V_{\alpha 2} \rightarrow V_{\beta}\otimes V_{\alpha 2}$

where each of the vector spaces are copies of $\mathbb{C}^2$
What is the statement of the problem?

There isn't any statement of the problem - it's something that I need for my Bachelor's thesis. However, I can provide some more detail. The operator $T$ is defined as $D_{\beta}R_{\beta\alpha n}\dots R_{\beta\alpha 2}R_{\beta\alpha 1}$, where the D and R operators are defined as above.

It's equation (44) on page 8 of this paper http://arxiv.org/pdf/1204.2089v3.pdf

Stephen Tashi
I want to write the product $D_{\beta}R_{\beta\alpha 1}R_{\beta\alpha 2}$, which makes sense on the space $V_{\beta} \otimes V_{\alpha 1} \otimes V_{\alpha 2}$.
Do you mean "I want to define a product..."?

I don't know what "makes sense" implies with respect to physics (because I don't understand the physics). Thinking only of mathematics, an attempt to refine the question could begin:

Let $V_{\beta},V_{\alpha_1},V_{\alpha_2}$ be vector spaces.
Let $R_{\beta \alpha_2}$ be an operator $R_{\beta \alpha_2}: V_\beta \otimes V_{\alpha_2} \rightarrow V_\beta \otimes V_{\alpha_2}$.

Construct an operator $R: V_{\beta} \otimes V_{\alpha_1} \otimes V_{\alpha_2} \rightarrow V_\beta \otimes V_{\alpha_1} \otimes V_{\alpha_2}$ such that ....
Roughly speaking, we want what $R$ does on $V_\beta \otimes V_{\alpha_1} \otimes V_{\alpha_2}$ to project to what $R_{\beta \alpha_2}$ does on $V_\beta \otimes V_{\alpha_2}$.

I don't know whether we are dealing with finite dimensional vectors spaces. Perhaps we are only dealing with linear operators. Perhaps you need to express the operator $R$ explicitly as a product of matrices or tensor product of operators.

In finite dimensions, you can find an embedding operation $E: V_\beta \otimes V_{\alpha_2} \rightarrow V_\beta \otimes V_{\alpha_1} \otimes V_{\alpha 2}$ and a projection operation $P: V_\beta \otimes V_{\alpha_1} \otimes V_{\alpha 2} \rightarrow V_\beta \otimes V_{\alpha_2}$ and define $R = E R_{\beta \alpha_2} P$

If the mathematical requirements aren't specific then I'd say this isn't a "homework type problem". You'd could ask about it in the physics sections of the forum.