Type-A path spaces and their matrix-unit tower

Let $\mathrm{\ell }\ge 3$. The path graph $A_{\mathrm{\ell }-1}$ has vertex set $\{0,1,\dots ,\mathrm{\ell }-2\}$ and an edge between $a$ and $b$ exactly when $\mathrm{\mid }a-b\mathrm{\mid }=1$. For $n\ge 0$ and a vertex $a$, let ${\mathsf{P}}_{n,a}$ be the set of sequences

with $a_0=0$, $a_n=a$, each $a_j\in \{0,1,\dots ,\mathrm{\ell }-2\}$, and $\mathrm{\mid }{a}_j-a_{j-1}\mathrm{\mid }=1$ for $1\le j\le n$. The set is finite; it may be empty. Let $V_{n,a}:={\mathbb{C}}^{({\mathsf{P}}_{n,a})}$ be the free complex vector space with basis $\{\mathrm{\mid }p⟩:p\in {\mathsf{P}}_{n,a}\}$, and put

omitting zero summands. For $p,q\in {\mathsf{P}}_{n,a}$, the matrix unit $E_{p,q}^{(n,a)}$ sends $\mathrm{\mid }q⟩$ to $\mathrm{\mid }p⟩$, sends every other path basis vector to zero, and is zero on every endpoint block other than $a$.

Equip each $V_{n,a}$ with the Hermitian inner product for which its path basis is orthonormal, and equip each endomorphism algebra with the corresponding adjoint. Thus $(E_{p,q}^{(n,a)}{)}^{\ast }=E_{q,p}^{(n,a)}$, and $K_n$ has the componentwise involution.

For $b\sim a$, write $pb=(a_0,\dots ,a_n,b)$ for the path obtained by appending $b$. Define the unital algebra map candidate ${\iota }_n:K_n\to K_{n+1}$ on matrix units by

and extend it linearly. The next local theorem proves that these assignments are injective unital $\ast$-homomorphisms and gives the corresponding branching rule. This is an abstract path-space tower definition; it does not assert that a particular quotient of a Temperley–Lieb algebra is isomorphic to $K_n$.

When $n=0$, the sole path is $(0)$ and $K_0\cong \mathbb{C}$. The vertex bounds are part of the definition, so a path attempting to step below $0$ or above $\mathrm{\ell }-2$ is absent rather than assigned a zero multiplicity by convention.