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I am from the math department and reading Belavin & Gebner's On the Algebraic Approach to Solvable Lattice Models.
I am trying to understand the left-hand side of Equation (2.2) on page 4.
What is meaning of the notation $\langle a_1, \ldots, a_n \mid X_i(u) \mid a_1', \ldots, a_n' \rangle$?
Physically, the $|a_1',...,a_n'⟩$ term represents a state of your system. Mathematically, it is a vector of the Hilbert space.
Each $a_i$ represents the occupation number of the $i$-th site of your lattice, ie. the number of particles on that site. Here, this notation would imply that your lattice has $n$ different sites, and that a state is fully characterized by the number of particles in each lattice site.
If you know second quantization, this state can also be written as $(b_1^\dagger)^{a_1'}...(b_n^\dagger)^{a_n'}|vac⟩$, where $|vac⟩$ is a state with no particles in the system at all, and each of the $b_i^\dagger$ operators adds one particle to the $i$-th site.
Then, $X_i(u)$ is an operator acting on your state, and $⟨a_1, ..., a_n|$ is also a state of the system but in the dual space.
One way to understand the full expression is the following. If your lattice starts from the state $|a_1',...,a_n'⟩$ and you apply the operator $X_i(u)$ on your system, you will end up with the projection of this new state of your system on the state $|a_1,...,a_n⟩$.
