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The fundamental quantum sl2 module in the Kauffman-bracket ribbon convention

Definition

Let $ω$ be an indeterminate, set $A = ω^{- 2}$ , and put $k^{'} = Q (ω)$ , $q_{K} = A^{2}$ , and $q_{L} = A^{- 2}$ . Work over the scalar extension to $k^{'}$ of the library Hopf algebra at $q_{L}$ . The parameter and Cartan-generator conversion from Korinman's convention is $q_{L} = q_{K}^{- 1}$ and $K_{L} = K_{K}^{- 1}$ ; the root generators $E, F$ correspond unchanged. The vector basis order remains $(v_{+}, v_{-})$ . Korinman displays these same column-action matrices for the root generators; no transpose is required. Thus the module matrices are $E = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}), F = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}), K_{K} = (\begin{array}{cc} q_{K} & 0 \\ 0 & q_{K}^{- 1} \end{array}) .$ Equivalently, for the library parameter use $K_{L} = K_{K}^{- 1} = diag (q_{L}, q_{L}^{- 1})$ and the library coproduct convention $Δ (E) = E \otimes K_{L}^{- 1} + 1 \otimes E$ , $Δ (F) = F \otimes 1 + K_{L} \otimes F$ . The parameters $q_{K}$ and $q_{L}$ are not identified.

The object $V$ is absolutely simple in the $k^{'}$ -linear representation category.

Korinman's §2.1 matrix $C_{K} = (\begin{array}{cc} 0 & ω \\ - ω^{5} & 0 \end{array})$ is labeled there as the matrix of his inverse half-ribbon operator on the underlying vector space $V$ . It is not the equivariant left-duality identification used below. Here $Ω$ denotes specifically the Snyder–Tingley operator $X$ , whose inverse matrix is computed independently below; it is not identified with $C_{K}$ .

To specify the half-ribbon action, use the Snyder–Tingley convention $q_{S T} = q_{K}$ , $K_{S T} = K_{K} = K_{L}^{- 1}$ , $E_{S T} = E$ , and $F_{S T} = F$ in the same ordered basis. Since $q_{K} = ω^{- 4}$ , choose $q_{S T}^{1 / 4} = ω^{- 1}$ . Let $X = J T_{w_{0}}$ be their half-ribbon element and let $Ω$ be its operator on $V$ . On the fundamental highest- and lowest-weight vectors, their formula gives $X v_{+} = - q_{S T}^{3 / 4} v_{-}$ and $X v_{-} = q_{S T}^{3 / 4} v_{+}$ . The half-ribbon action therefore gives the actual inverse matrix and its square: ${Mat}_{(v_{+}, v_{-})} (Ω^{- 1}) = (\begin{array}{cc} 0 & - ω^{3} \\ ω^{3} & 0 \end{array}), Ω^{2} = - ω^{- 6} {id}_{V} = - A^{3} {id}_{V} .$ Snyder–Tingley’s associated Hopf-algebra ribbon element is $X^{- 2}$ ; the positive categorical twist in this Kauffman-bracket convention is its inverse $X^{2} = Ω^{2}$ . In the ordered basis $(v_{+} \otimes v_{+}, v_{+} \otimes v_{-}, v_{-} \otimes v_{+}, v_{-} \otimes v_{-})$ , the positive braiding is $c_{V, V} = (\begin{array}{cccc} A & 0 & 0 & 0 \\ 0 & 0 & A^{- 1} & 0 \\ 0 & A^{- 1} & A - A^{- 3} & 0 \\ 0 & 0 & 0 & A \end{array}) .$ Its ribbon quantum dimension is $δ = - A^{2} - A^{- 2}$ . The half-ribbon and universal $R$ -matrix are those of the topological half-ribbon representation category; the selected ribbon element is the Kauffman-bracket one, not the usual $U_{q} ({s l}_{2})$ ribbon element.

Let $v_{+}^{*}, v_{-}^{*}$ be the ordinary dual basis. The left-dual action on $V^{*}$ is the contragredient action $(x \cdot f) (v) = f (S (x) v)$ for the antipode $S$ of the stated library Hopf convention. Identify the displayed copy of $V$ with this left dual $V^{\lor}$ by $v_{+} \mapsto v_{-}^{*}$ and $v_{-} \mapsto - A^{- 2} v_{+}^{*}$ . In this identification the evaluation and coevaluation maps are $B (v_{+} \otimes v_{-}) = 1, B (v_{-} \otimes v_{+}) = - A^{- 2}, B (v_{+} \otimes v_{+}) = B (v_{-} \otimes v_{-}) = 0,$ $U (1) = - A^{2} v_{+} \otimes v_{-} + v_{-} \otimes v_{+} .$ These are the explicit duality maps used in the closure trace.

The positive Artin generator is drawn with braid height increasing upward; the left input strand ends on the right, and the right input strand passes over it:

The uninterrupted strand in the drawing is the overpassing strand. This is the crossing convention for the displayed matrix; the proof below uses its matrix entries and the half-ribbon source formulas, not the drawing.

Definition references

Drinfeld–Jimbo algebra presentation

Library coproduct convention

Absolute simplicity

Braiding definition

Ribbon twist definition

Left-dual maps and snake identities

Korinman, Finite presentations for stated skein algebras and lattice gauge field theory, §2.1

Snyder and Tingley, The half-twist for $U_{q} (g)$ representations, §§3–5

Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§8.3, 8.9–8.10

Facts & Assumptions

Given: The generic parameter $A$ , the basis $(v_{+}, v_{-})$ , and the two explicitly distinguished parameters $q_{K} = A^{2}$ and $q_{L} = A^{- 2}$ .

[F1]

The Drinfeld–Jimbo presentation has $K E K^{- 1} = q^{2} E$ , $K F K^{- 1} = q^{- 2} F$ , and $E F - F E = (K - K^{- 1}) / (q - q^{- 1})$ in the ${s l}_{2}$ case (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F2]

In the library convention, $Δ (E) = E \otimes K^{- 1} + 1 \otimes E$ , $Δ (F) = F \otimes 1 + K \otimes F$ , and the antipode is $S (E) = - E K$ , $S (F) = - K^{- 1} F$ , $S (K) = K^{- 1}$ (The Drinfeld–Jimbo formulas define a Hopf algebra).

[F3]

A braiding is a natural isomorphism $c_{X, Y} : X \otimes Y \to Y \otimes X$ satisfying the two hexagon identities (Braiding).

[F4]

A ribbon twist is a natural automorphism satisfying $θ_{X \otimes Y} = (θ_{X} \otimes θ_{Y}) \circ c_{Y, X} \circ c_{X, Y}$ and compatible with duality (Twist and ribbon structure).

[F5]

Korinman's full-text §2.1 gives the generic two-dimensional module, the displayed matrix of its inverse half-ribbon operator $C_{K}$ , the positive braiding matrix, and the Kauffman-bracket quantum dimension $- q_{K} - q_{K}^{- 1}$ ; the parameter, Cartan-generator, and column-action conversions used here are stated in the Definition (Korinman, §2.1).

[F6]

Snyder–Tingley Lemma 3.10 computes the action of $X = J T_{w_{0}}$ on highest- and lowest-weight vectors; Theorems 3.11 and 5.2 establish its standard braiding and half-ribbon structure, with associated Hopf-algebra ribbon element $v = X^{- 2}$ satisfying $Δ (v) = (v \otimes v) (R_{21} R)^{- 1}$ (Snyder–Tingley, §§3, 5).

[F7]

Absolute simplicity means $End (V) = k^{'} {id}_{V}$ (Absolutely simple objects).

[F8]

A left dual has evaluation $V^{\lor} \otimes V \to 1$ and coevaluation $1 \to V \otimes V^{\lor}$ satisfying both snake identities (Left dual and right dual object).

Proof

technique · direct

Given: The data in the Definition.

1.1F1F2givenalgebra

The displayed $E, F, K_{K}$ satisfy the ${s l}_{2}$ relations: conjugating $E$ by $K_{K}$ multiplies its only nonzero entry by $q_{K}^{2}$ , conjugating $F$ multiplies its only nonzero entry by $q_{K}^{- 2}$ , and $E F - F E = diag (1, - 1) = (K_{K} - K_{K}^{- 1}) / (q_{K} - q_{K}^{- 1})$ , with nonzero denominator since $q_{K} - q_{K}^{- 1} = ω^{- 4} - ω^{4} \neq 0$ in $k^{'}$ . Replacing $(q_{K}, K_{K})$ by $(q_{L}, K_{L}) = (q_{K}^{- 1}, K_{K}^{- 1})$ leaves the commutator quotient unchanged and gives the stated library toral relations. The coproduct conversion in the Definition is exactly [F2] after this substitution.

1.2F7givenalgebra

Any $U_{q_{K}} ({s l}_{2})$ -endomorphism of $V$ commutes with $K_{K}$ , whose distinct eigenvalues force it to be diagonal in $(v_{+}, v_{-})$ . Commuting with $E (v_{-}) = v_{+}$ forces the two diagonal entries to agree. Thus $End (V) = k^{'} {id}_{V}$ , which is [F7]'s absolute-simplicity condition.

1.3F3F5F6givenalgebra

In Korinman's convention the positive braiding is $P D (1 + (q_{K} - q_{K}^{- 1}) E \otimes F)$ , where $P$ is the flip and $D = diag (A, A^{- 1}, A^{- 1}, A)$ in the displayed tensor basis. Since $E^{2} = F^{2} = 0$ , $(E \otimes F)^{2} = 0$ and the source $q$ -exponential truncates after its linear term. Applying this operator to the four basis vectors gives, in order, $A v_{+} \otimes v_{+}$ , $A^{- 1} v_{-} \otimes v_{+}$ , $A^{- 1} v_{+} \otimes v_{-} + (A - A^{- 3}) v_{-} \otimes v_{+}$ , and $A v_{-} \otimes v_{-}$ . These are exactly the columns of the displayed matrix; [F6] supplies the half-ribbon completion in which this finite action is a braiding.

1.4F2F5F8givenalgebra

The bilinear map $B$ and vector $U (1)$ are $U_{q_{K}} ({s l}_{2})$ intertwiners. The $K_{K}$ -weights in each nonzero pairing and coevaluation term multiply to $1$ . For the only nonzero $E$ -pairing, $Δ (E) (v_{-} \otimes v_{-}) = q_{K}^{- 1} v_{+} \otimes v_{-} + v_{-} \otimes v_{+}$ , whose $B$ -value is $q_{K}^{- 1} - q_{K}^{- 1} = 0$ ; the other basis inputs give zero or a multiple of $v_{+} \otimes v_{+}$ , also killed by $B$ . For $F$ , the only nonzero check is $Δ (F) (v_{+} \otimes v_{+}) = v_{-} \otimes v_{+} + q_{K}^{- 1} v_{+} \otimes v_{-}$ , whose $B$ -value is $- q_{K}^{- 1} + q_{K}^{- 1} = 0$ . On coevaluation, $Δ (E)$ sends $- q_{K} v_{+} \otimes v_{-} + v_{-} \otimes v_{+}$ to $- q_{K} v_{+} \otimes v_{+} + q_{K} v_{+} \otimes v_{+} = 0$ , and $Δ (F)$ sends it to $- q_{K} v_{-} \otimes v_{-} + q_{K} v_{-} \otimes v_{-} = 0$ ; the $K_{K}$ -weight is zero. Since these checks hold on the generators and $Δ, ε$ are algebra maps, they establish invariance of $B$ and $U$ for every algebra element. The Hopf identity $\sum x_{(1)} \otimes x_{(2)} S (x_{(3)}) = x \otimes 1$ then gives $B (x v, w) = \sum B (x_{(1)} v, x_{(2)} S (x_{(3)}) w) = \sum ε (x_{(1)}) B (v, S (x_{(2)}) w) = B (v, S (x) w)$ . Thus $ϕ (v) = B (v, -)$ intertwines the stated contragredient action. Its matrix from $V$ to $V^{*}$ has nonzero determinant $A^{- 2}$ , so it is an isomorphism. Transporting the standard coevaluation $1 \mapsto v_{+} \otimes v_{+}^{*} + v_{-} \otimes v_{-}^{*}$ by $ϕ^{- 1}$ gives exactly the displayed $U$ . The snake composites are identities: $(B \otimes 1) (1 \otimes U)$ returns $v_{+}$ and $v_{-}$ with coefficients $B (v_{+}, v_{-}) = 1$ and $(- A^{2}) B (v_{-}, v_{+}) = 1$ , respectively, while $(1 \otimes B) (U \otimes 1)$ returns them with coefficients $(- A^{2}) B (v_{-}, v_{+}) = 1$ and $B (v_{+}, v_{-}) = 1$ . By [F8], these are the defining left-dual identities. The closed loop is $B (U (1)) = - A^{2} - A^{- 2} = δ$ . Thus these maps give the stated self-duality and quantum dimension.

2.1F4F6step 1.3algebra∎

For ${s l}_{2}$ , the fundamental highest weight $λ$ satisfies $(λ, λ) = 1 / 2$ , $(λ, ρ) = 1 / 2$ , and $⟨ 2 λ, ρ^{\lor} ⟩ = 1$ . Snyder–Tingley’s formula [F6] therefore gives $X v_{+} = - q_{S T}^{3 / 4} v_{-}$ and $X v_{-} = q_{S T}^{3 / 4} v_{+}$ . With $q_{S T}^{1 / 4} = ω^{- 1}$ , this yields $Mat (Ω) = (\begin{array}{cc} 0 & ω^{- 3} \\ - ω^{- 3} & 0 \end{array})$ and $Mat (Ω^{- 1}) = (\begin{array}{cc} 0 & - ω^{3} \\ ω^{3} & 0 \end{array})$ , whose square is $Ω^{2} = - A^{3} I$ . The Hopf ribbon identity in [F6] gives $Δ (v^{- 1}) = (v^{- 1} \otimes v^{- 1}) R_{21} R$ ; since the monodromy $R_{21} R$ is $c_{Y, X} c_{X, Y}$ , this inverse acts as the categorical twist for the balancing convention in [F4]. Thus the positive categorical twist is $v^{- 1} = X^{2} = Ω^{2}$ .

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Sources

Julien Korinman, Finite presentations for stated skein algebras and lattice gauge field theory, Algebraic & Geometric Topology 23 (2023), §2.1, subsection “Relation with Uqsl2 and Oq[SL2]”, printed pp. 1256–1257 (PDF pp. 10–11) (standard reference, not scraped)
Noah Snyder and Peter Tingley, The half-twist for U_q(g) representations, Algebra & Number Theory 3 (2009), §§3–5, printed pp. 810–830 (standard reference, not scraped)
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, §8.3.13 and §§8.9–8.10 (standard reference, not scraped)