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I'm working on quantum mechanics of linear $SU(N)$ chain of sites.

Specifically, I would like to study a Tensor product of $m$ Lie algebras $\mathfrak{s}\mathfrak{u}(N)$

$$ V\in SU(N)\\ W = V\otimes^m \subset SU(N^m) $$ Has such a study been undertaken before ? How would one approach this topic ? If one assumes standard knowledge of representation theory/group theory.

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