The Tensor Algebra

Here, we drop one last key bit of algebraic machinery, necessary before we can kick off our description of ring-switching. The goal is to define the tensor algebra, an algebraic object that plays an essential role in the technique.

Throughout our treatment thus far, we have often purposefully played on both sides of a divide: the formal–algebraic divide, let's say. (This motif traces back to the beginning). On the one hand, we have formal manipulations of lists and bits (things like splitting and concatenating). On the other hand, these operations simultaneously correspond to algebraic operations (like basis-combination and decomposition).

Our treatment of the tensor algebra will follow this trend. First, we will describe it in rather surface-level terms: just to keep things down-to-earth. On the other hand, there is always an algebraic story running in parallel with the formal one. That latter story will become essential when it comes time to evaluate.

First Description

As usual, we are going to fix two tower levels $\mathcal{T}_\iota \subset \mathcal{T}_\tau$. We write $\kappa \coloneqq \tau - \ell$; we recall that $\mathcal{T}_\tau$ is a $2^\kappa$-dimensional vector space over $\mathcal{T}_\tau$, and that we have an association between $\mathcal{T}_\tau$-elements $\alpha$ and lists of $\mathcal{T}_\iota$-elements $(\alpha_0, \ldots , \alpha_{2^\kappa - 1})$.

In other words, $\mathcal{T}_\tau$-elements and lists of $2^\kappa$ $\mathcal{T}_\iota$-elements are "the same thing".

What if we make this idea two-dimensional? Nothing stops us from defining a square, $2^\kappa \times 2^\kappa$ array, each of whose cells is a $\mathcal{T}_\iota$-element. Let's write $a$ for just such an array.

The motivation thus far is lacking, but for starters, we can make a few points. First, by looking at $a$'s columns—and interpreting each one as a $\mathcal{T}_\iota$-element—we can "cast" $a$ as an array of $2^\kappa$ $\mathcal{T}_\tau$-elements $\left( a_v \right)_{v \in \mathcal{B}_\kappa}$. Similarly, we can do an identical thing for $a$'s rows, yielding a different decomposition $\left( a_u \right)_{u \in \mathcal{B}_\kappa}$. Thus, the square array $a$ has "two different decompositions at once". These will end up relating to each other in interesting ways.

Similarly, we can define the multiplication of $a$ by some element $\alpha \in \mathcal{T}_\tau$ in two different ways. For one, we can multiply each column of $a$ by $\alpha$; this will give us the column decomposition $\left( a \cdot a_v \right)_{v \in \mathcal{B}_\kappa}$. We can also multiply each row of $a$ by $\alpha$, yielding $\left( a \cdot a_u \right)_{u \in \mathcal{B}_\kappa}$.

The Algebraic Side

As it happens, this object is nothing other than an algebraic one: namely, the $\mathcal{T}_\iota$-algebra $A \coloneqq \mathcal{T}_\tau \otimes_{\mathcal{T}_\iota} \mathcal{T}_\tau$. That is, it's the tensor product of $\mathcal{T}_\tau$ with itself over its own subfield $\mathcal{T}_\iota$. The whole thing takes place in the category of $\mathcal{T}_\iota$-algebras. The column and row decompositions above fall out as basic properties of the tensor product, as do the multiplication operations. This entire setup is described in [DP24, § 2.5].

Two particular algebraic operations will prove especially important. Given a single $\mathcal{T}_\tau$-element, we can always view it as an $A$-element, by inscribing it (in $\mathcal{T}_\iota$-coordinates) into the leftmost column of a $2^\kappa \times 2^\kappa$ array (and putting 0s elsewhere). This gives us the column embedding $\varphi_0 : \mathcal{T}_\tau \to A$. Symmetrically, we can equally embed any $\mathcal{T}_\tau$-element into the top row; this gives a second embedding $\varphi_1 : \mathcal{T}_\tau \to A$.

The key point is that these operations are algebraic and mathematically meaning, and not just formal; this fact will become crucial in what follows.