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The type-A quantum algebra and its tensor-action formulas

Definition

Let $F = Q (q)$ with $q$ an indeterminate, let $N \geq 1$ , and put $K_{i} = L_{i} L_{i + 1}^{- 1}$ for $1 \leq i < N$ . Start with the free associative $F$ -algebra on $L_{j}, L_{j}^{- 1}$ ( $1 \leq j \leq N$ ) and $E_{i}, F_{i}$ ( $1 \leq i < N$ ), and quotient by the two-sided ideal generated by the following relations:

$L_{j} L_{j}^{- 1} = L_{j}^{- 1} L_{j} = 1$ and $L_{j} L_{k} = L_{k} L_{j}$ for all $j, k$ .
$L_{j} E_{i} L_{j}^{- 1} = q^{δ_{j, i} - δ_{j, i + 1}} E_{i}$ and $L_{j} F_{i} L_{j}^{- 1} = q^{- δ_{j, i} + δ_{j, i + 1}} F_{i}$ .
$[E_{i}, F_{j}] = δ_{i j} (K_{i} - K_{i}^{- 1}) / (q - q^{- 1})$ .
$[E_{i}, E_{j}] = [F_{i}, F_{j}] = 0$ when $∣ i - j ∣ > 1$ .
For $∣ i - j ∣ = 1$ , $E_{i}^{2} E_{j} - (q + q^{- 1}) E_{i} E_{j} E_{i} + E_{j} E_{i}^{2} = 0$ and $F_{i}^{2} F_{j} - (q + q^{- 1}) F_{i} F_{j} F_{i} + F_{j} F_{i}^{2} = 0$ .

Here $[x, y] = x y - y x$ . The free algebra and quotient construction are those of The free associative R-algebra on a set and descent of relations; this definition asserts existence of the presented associative algebra, not the well-definedness of structural maps on its quotient.

On $V_{N}$ with ordered basis $e_{1}, \dots, e_{N}$ , prescribe the candidate matrices $L_{j} e_{a} = q^{δ_{j a}} e_{a}, E_{i} e_{a} = δ_{a, i + 1} e_{i}, F_{i} e_{a} = δ_{a, i} e_{i + 1} .$ Prescribe candidate coproduct and counit values by $Δ (L_{j}) = L_{j} \otimes L_{j}, Δ (E_{i}) = E_{i} \otimes K_{i} + 1 \otimes E_{i}, Δ (F_{i}) = F_{i} \otimes 1 + K_{i}^{- 1} \otimes F_{i},$ and $ϵ (L_{j}^{\pm 1}) = 1$ , $ϵ (E_{i}) = ϵ (F_{i}) = 0$ . The following relation-descent lemma proves that these assignments define the vector representation and bialgebra structure, and hence the tensor actions used below.

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Sources

Pavel Etingof and Mykola Semenyakin, A brief introduction to quantum groups (standard reference, not scraped)