# Implementing the tensor algebra functor

**URL:** <https://discourse.julialang.org/t/implementing-the-tensor-algebra-functor/64433>\
**Category:** Numerics\
**Tags:** question, package, linearalgebra, tensors, functors\
**Created:** [July 10, 2021, 5:32pm UTC](https://discourse.julialang.org/t/implementing-the-tensor-algebra-functor/64433 "2021-07-10T17:32:28Z")\
**Posts on this page:** 1\
**Page:** 1

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**Author:** ![choquet](https://avatars.discourse-cdn.com/v4/letter/c/f08c70/32.png) [@choquet](https://discourse.julialang.org/u/choquet)\
**Post date:** [July 10, 2021, 5:32pm UTC](https://discourse.julialang.org/t/implementing-the-tensor-algebra-functor/64433/1 "2021-07-10T17:32:28Z")

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I’m trying to implement the free tensor algebra functor T: given a vector space V it creates another vector space T(V). This vector space T(V) has many nice properties, for example, T(V) is not just a vector space but also forms an algebra. It is used in many constructions; see [https://en.wikipedia.org/wiki/Tensor\_algebra](https://en.wikipedia.org/wiki/Tensor_algebra) for details.

What is nice is that it can be realized as a direct sum of tensor products of V. In fact, I’m only interested in the case when V is R^d and I’m only interested in the first k entries in the direct sum that forms T(V).

What matters is that the basic operations, addition and multiplication of elements in T(V), are carried out quickly. I had a look at the various tensor packages available but to be honest I’m a bit overhelmed with the choice and some of them look like they are no longer maintained. Any recommendations would be appreciated.