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2016-05-07 05:27:26 -0500 commented answer Confused about FreeAlgebra quotients

letterplace is a completely different implementation of FreeAlgebras which emulates noncommutative homogeneous polynomials as commutative polynomials using Singular. It sure works well, but it has some limitations, for instance the elements of the FreeAlgebra aren't callable and hence substitutions won't work. On the other hand, if the described behaviour of the non letterplace implementation is not a bug, then one might ast what are such quotients good for? In any case, it is desirable that both implementations maintained a good compatibility level with each other.

2016-05-07 05:27:26 -0500 asked a question Exterior and tensor product of algebras


I've been trying to construct something like $\Lambda^2 A \otimes A$ where $A$ is a free associative $\mathbb{Q}$-algebra on some generators, say, $x$ and $y$, without success. I think I understand the basics of the FreeAlgebra object in Sage, but I could not find more specific information about this kind of construction in the documentation or here in this forum. Ultimately, I would like to form and calculate with expressions like $x y \wedge xx \otimes xy$ where $x,y\in \mathbb{Q}\langle x,y\rangle$, the free associative unital algebra, and such that, for instance, $xy \wedge xx \otimes xy = - xx \wedge xy \otimes xy$. Does anyone know whether this is or is not possible to achieve in Sage at the moment? Any help will be greatly appreciated.