$\mathfrak{sl}_2$, Verlinde, and dihedral models

Generic $\mathfrak{sl}_2$ representation rules

The generic semisimple representation theory of $U_q(\mathfrak{sl}_2)$ is the infinite precursor of the finite $\mathrm{SU}(2)$ Verlinde categories below. This constructor realizes its Clebsch–Gordan rules in the Kauffman–Lins trivalent recoupling basis; it does not store matrices for a quantum-group action.

sl2_representations(K,q) constructs a skeletal model of the fusion rules and recoupling coefficients. It has infinitely many simple labels and is therefore not a fusion category. Under the usual generic semisimplicity assumptions its simple labels are $V_0,V_1,\ldots$, with

\[\label{eq:uqsl2-generic-fusion} V_i\otimes V_j=\bigoplus_{r=0}^{\min(i,j)}V_{i+j-2r}.\]

This is the quantum Clebsch–Gordan rule of Huang (2017), Proposition 5.7; see also Kassel (1995), Chapter VII, §7, pp. 157–162. For this categorical interpretation, take $K$ to be a field of characteristic zero and $q\in K^\times$ to be either $1$ or generic, rather than a root of unity. All quantum integers and theta-net denominators used by a requested recoupling block must be nonzero. This is not a construction of the full representation category at roots of unity or in positive characteristic. The constructor does not validate these restrictions.

The call sl2_representations(K) uses $q=1$, while the zero-argument call uses the algebraic complex numbers and $q=1$.

The current model supports finite direct sums, $\operatorname{Hom}$ spaces, tensor products of nonzero objects, and recoupling associators. It does not implement duals, evaluation and coevaluation, a braiding, or pivotal and spherical structures. Those rigid operations remain unavailable despite the current is_tensor(C) declaration.

An object stores a sparse vector of simple multiplicities. A morphism stores one matrix block for each represented simple label, with rows indexed by source copies and columns by target copies; matrix(f) forms their block diagonal matrix. Tensor products of blocks use kronecker_product, with the coordinate from the right factor varying fastest. Object equality compares the parent and the occupied multiplicities; morphism equality also compares endpoints and occupied blocks. When constructing a morphism from an array or dictionary of blocks, supply compatible indices, sizes, coefficient fields, and endpoints; the low-level constructor assumes these conditions.

The argument q fixes the quantum integers through $[2]=q+q^{-1}$. For external labels $i,j,k$, output label $w$, ascending left channel $m$, and ascending right channel $n$, the stored entry in the row indexed by $m$ and column indexed by $n$ is

\[\label{eq:uqsl2-kauffman-lins-symbol} \left\{\begin{matrix}j&i&n\\w&k&m\end{matrix}\right\}_{\!\mathrm{KL}}.\]

This is exactly the recoupling coefficient defined in Kauffman and Lins (1994), Chapter 9, §§9.10–9.12, pp. 93–101, with the argument order shown above.

Here C[i] denotes $V_i$ with its mathematical, zero-based label; this is an exception to indexing by position in simples(C) for finite skeletal categories.

using TensorCategories, Oscar
C = sl2_representations(QQ,QQ(1))
@assert C[1]⊗C[1] == C[0]⊕C[2]
decompose(C[1]⊗C[1])

2-element Vector{Tuple{TensorCategories.UqSl2rep, Int64}}:
 (V0, 1)
 (V2, 1)

Use simples(C,n) for the first $n$ simple objects. Algorithms that require a finite set of simples do not apply to this category.

Verlinde modular categories

The $\mathrm{SU}(2)$ Verlinde modular categories are root-of-unity semisimplifications that occur in Wess–Zumino–Witten conformal field theory and in $\mathrm{SU}(2)$ Chern–Simons anyon models. This constructor directly installs the resulting skeletal fusion rules and Temperley–Lieb recoupling coefficients; it does not construct the semisimplification from quantum-group modules.

For an integer $m\geq 0$, verlinde_category(m) has rank $m+1$, with labels $X_0,\ldots,X_m$. The default field is $\mathbb Q(\zeta_{4m+8})$ and the implementation first chooses a root $z$ of order $2m+4$. Julia C[1] is $X_0$ here. The fusion rule is

\[\label{eq:uqsl2-level-fusion} X_a\otimes X_b= \bigoplus_{\substack{c=|a-b|\\c\equiv a+b\;(\mathrm{mod}\,2)}}^{ \min(a+b,\,2m-a-b)} X_c.\]

It is the level-$m$ $\mathrm{SU}(2)$ fusion rule: equation (1) of Taylor and Woodward (2006) becomes the displayed formula after setting $r=m+2$ and replacing each half-integer label $j$ by $2j$.

The full positional call is verlinde_category(K,m,l,t); omitting the last two arguments uses $l=t=1$. The cyclotomic-field overload is verlinde_category(m,l,t), again with $l=t=1$ by default. For these default values, let $A$ be the bracket parameter of Kauffman and Lins (1994), Chapters 6–8, pp. 45–92. The parameters are related by

\[\label{eq:uqsl2-kauffman-lins-parameters} r=m+2,\qquad z=A^2,\qquad w=\sqrt z=A, \qquad [n]=\frac{z^n-z^{-n}}{z-z^{-1}}.\]

The stored coefficients use the Temperley–Lieb recoupling coefficient

\[\label{eq:uqsl2-kauffman-lins-formula} \left\{\begin{matrix}a&b&i\\c&d&j\end{matrix}\right\}_{\!\mathrm{KL}} =\frac{\operatorname{Tet}[a,b,i;c,d,j] \, (-1)^i[i+1]} {\Theta(a,d,i)\Theta(b,c,i)}\]

in the trivalent basis of Kauffman and Lins (1994), Chapter 9, pp. 93–101. Here $\Theta$ and $\operatorname{Tet}$ denote the theta-net and tetrahedral-net evaluations defined in Kauffman and Lins (1994), §§9.10–9.12, pp. 97–99. Consequently, for external labels $a,b,c$, output label $d$, ascending left channel $e$, and ascending right channel $f$, the entry in row $e$ and column $f$ of the stored associator block is

\[\label{eq:uqsl2-package-symbol-permutation} \left\{\begin{matrix}b&a&f\\d&c&e\end{matrix}\right\}_{\!\mathrm{KL}}.\]

For general $l$, the quantum integers are instead generated from $z^l+z^{-l}$. The cited Kauffman–Lins presentation identifies the default $l=1$; the other $l$-values are additional parameter choices whose literature normalization is unspecified. For the usual semisimple level-$m$ model one must have

\[\label{eq:uqsl2-coprimality} \gcd(l,m+2)=1.\]

Equivalently, the quantum integers $[1],\ldots,[m+1]$ are then all nonzero. Without this condition, a denominator in the recoupling formulas can vanish. The constructor does not enforce the condition.

The braiding is supplied when the field contains a square root $w$ of $z$:

\[\label{eq:uqsl2-r-symbol} R^{ab}_c=(-1)^{(a+b-c)/2}(w^t)^{(a(a+2)+b(b+2)-c(c+2))/2}.\]

For $t=1$ this is the local braiding of Kauffman and Lomonaco Jr. (2010), §10, Figure 39, with the crossing orientation shown there. The other square root or the inverse crossing changes the literal scalar. The literal braiding comparison is likewise for $t=1$. Not every choice of $l,t$ gives compatible associators and braiding. The constructor does not check the pentagon or hexagon identities. It also retains the skeletal initializer's all-one pivotal components without checking the pivotal or spherical identities.

using TensorCategories, Oscar
C = verlinde_category(2)
@assert length(simples(C)) == 3
@assert is_isomorphic(C[2]⊗C[2], C[1]⊕C[3])[1]
@assert pentagon_axiom(C)
@assert hexagon_axiom(C)
fpdim.(simples(C))

3-element Vector{QQBarFieldElem}:
 {a1: 1.00000}
 {a2: 1.41421}
 {a1: 1.00000}

Associators and braiding blocks are generated lazily. Use associator, braiding, or six_j_symbols rather than reading uninitialized entries of C.ass immediately after construction.

Dihedral Temperley–Lieb models

Two-colored Temperley–Lieb categories encode the finite dihedral case of Soergel calculus. The package supplies a two-summand-unit skeleton and its simple-unit even sector.

For the finite dihedral parameter $m\geq 3$, I2(m) has $2(m-1)$ simple labels, corresponding to alternating words beginning with $s$ or $t$. Its unit has two summands. The constructors do not validate this range. I2subcategory(m) retains $\lfloor m/2\rfloor$ labels $B_s,B_{sts},\ldots$, with simple unit $B_s$. The default coefficient field is $\mathbb Q(\zeta_{2m})$. To supply another coefficient field, use the constructors I2(m,K) and I2subcategory(m,K), with the field as the second argument. The supplied field must contain a primitive $2m$-th root of unity and the inverses required by the recoupling formulas.

Their labels and fusion rules model the two-colored Temperley–Lieb category at a root of order $2m$, modulo the negligible part, and their associator blocks use Kauffman–Lins recoupling coefficients. The two colors, alternating labels, and relation with finite dihedral Soergel calculus are described in Elias (2016), §1.4 and Proposition 1.2, pp. 6–7; §4.3, Proposition 4.11 and Remark 4.12, p. 30; §6.4, Theorem 6.29, p. 61.

The fusion rules of I2subcategory(m) are those of the even, or adjoint, sector $\mathrm{SO}(3)_{m-2}$ of the level-$(m-2)$ $\mathrm{SU}(2)$ category. This parameter shift is also the one in the identification of the middle dihedral asymptotic category with $\mathrm{SO}(3)_{m-2}$ (Mackaay et al., 2023; §8(c), equation (8.4)).

Both constructors use the Kauffman–Lins recoupling matrices and neither supplies a braiding. I2subcategory explicitly installs all-one pivotal components. The full I2 model retains default all-one components from the skeletal initializer, but they are not validated pivotal data. The literature identification covers the fusion rules and Kauffman–Lins recoupling formulas; no basis-by-basis equivalence with the cited diagrammatic category, or identification of the all-one components with its pivotal structure, is fixed. Evaluation and coevaluation in the full I2 model are incomplete, so it does not currently supply a complete rigid structure.