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I've been googling the definition of it, and it seems like somehow it's related to a polynomial ring.

But I still quite don't get it.

Is a free algebra just a free group with additional operation (which is multiplication)?

Can anyone give me a simple explanation of it or any reference to it?

Thanks.

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1 Answer

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A free (associative) algebra is the non commutative analogue of the (associative) algebra of polynomials over a ring. If we have a ring $R$ we can get the polynomials over this ring in $R[X_1, X_2, \cdots,X_n]$. These polynomials have $n$ letters. Each term in a polynomial element could be thought of as a list of elements with their powers and some unit: $aX_1^{p_1}X_2^{p_2}\cdots X_n^{p_n}$. Ultimately the point is that order does not matter.

For a free algebra, the idea is to do the same but now we take care with order. Thus we have an algebra over some ring $R$ whose basis is all $n$ letter words. A typical element may look like this: $X_1X_3^2X_2$.

It could almost be thought of as the polynomial algebra is the free algebra modulo some commutator relation. The point is to not be able to change the order of multiplication as you could before.

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