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I will risk myself on a general/vague but nonetheless "titillating" question:

-One may have encountered the fact that any representation of a compact group can be written as a direct sum of irreducible representations, in particular the tensor product of two representations i.e. we have an equality of representations of the form $$V_a \otimes V_b = \bigoplus_{i\in I} V_i $$ -Similarly to the situation for integers, one has for a vector space $V$ over $\mathbb{K}$ $$ V\otimes K^{n}= \bigoplus_{i=1}^n V \quad\text{(to compare with)}\; d\times n = d+d+\cdots+ d $$

Is there a category (moinoidal abelian?, for which at least the direct sum and the tensor product exist) where under some appropriate conditions one has equality of a non trivial tensor product and a non trivial direct sum?

Edit: Indeed I'm not so much interested in the unicity of the decomposition. However I find it incredible that in the two examples above, one has an object that satisfies two distinct universal properties, namely the one for a tensor product and the one for a coproduct.

In the case one has the direct sum of only itself, the universal doesn't impose anything.

Such situation where two different universal properties are related or actually coincide does occur, e.g. of preadditive category where product and coproduct agree (from biproduct in wikipedia).

Noix07
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  • Relevant but complicated: http://math.stackexchange.com/questions/291644/alternative-construction-of-the-tensor-product-or-pass-this-secret?rq=1 – Noix07 Jan 21 '14 at 14:37
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    Your question might need additional clarity - every object is the direct sum of just itself. In the Lie group setting, at least when the group is simple, the category of representations is Krull-Schmidt, so every object is a direct sum of indecomposables, rather than this being something special about tensor products. Same for the category of finite dimensional vector spaces (although of course what you have written is not in general the decomposition into a sum of indecomposables). – mdp Jan 21 '14 at 14:37
  • The question is unclear, therefore -1. – Martin Brandenburg Jan 21 '14 at 14:53
  • Question are many times unclear. It is of the kind, when does such or such property hold. If I knew precisely the conditions, then I would write a theorem and not ask a question – Noix07 Jan 21 '14 at 15:05

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In a commutative monoid $G\ni g$ one has $n\cdot g = g+ \cdots + g$ n times, for n integer.

i.e. there is a natural action of the non negative intergers on any commutative monoid. Multiplication is the action of the integer, addition is the operation in the monoid.

It seems to transpose to monoidal categories with direct sum.

  • For the first example, in the so called semi simple category, any object can be written as a finite direct sum. cf. nLab
  • The second example is of the form $$ V \otimes (W\oplus Z) = (V\otimes W)\oplus(V\otimes Z)$$ Moreover the field $\mathbb{K}$ is a "tensor unit" or left/right unitor in wiki
  • cf. also "isotypical representation" (of a compact group I guess), "6 j symbol", "fusion rule"
  • check this question
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Noix07
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