Chunrui Ai, Chongying Dong, Xingjun Lin
In this paper, extensions of nonunitary rational Virasoro vertex operator algebras corresponding to some exceptional modular invariants are constructed. The uniqueness of these extensions is also established.
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In this paper, we study Virasoro vertex algebras and affine vertex algebras over a general field of characteristic $p>2$. More specifically, we study certain quotients of the universal Virasoro and affine vertex algebras by ideals related to the $p$-centers of the Virasoro algebra and affine Lie algebras. Among the main results, we classify their irreducible $\mathbb{N}$-graded modules by explicitly determining their Zhu algebras and show that these vertex algebras have only finitely many irreducible $\mathbb{N}$-graded modules and they are $C_2$-cofinite.
In this paper, we study a new kind of vertex operator algebra related to the twisted Heisenberg-Virasoro algebra, which we call the twisted Heisenberg-Virasoro vertex operator algebra, and its modules. Specifically, we present some results concerning the relationship between the restricted module categories of twisted Heisenberg-Virasoro algebras of rank one and rank two and several different kinds of module categories of their corresponding vertex algebras. We also study fully the structures of the twisted Heisenberg-Virasoro vertex operator algebra, give a characterization of it as a tensor …
We give an abstract construction, based on the Belavin-Polyakov-Zamolodchikov equations, of a family of vertex operator algebras of rank $26$ associated to the modified regular representations of the Virasoro algebra. The vertex operators are obtained from the tensor products of intertwining operators for a pair of Virasoro algebras. We explicitly determine the structure coefficients that yield the axioms of VOAs. In the process of our construction, we obtain new hypergeometric identities.
We consider all genus two correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of a sequence of globally defined differential operators on which the genus two Siegel modular group $Sp(4,\mathbb{Z})$ has a natural action.
We consider all genus zero and genus one correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of graph theory related to derangements in the genus zero case and to partial permutations in the genus one case.
In this article, we show that a framed vertex operator algebra V satisfying the conditions: (1) V is holomorphic (i.e., V is the only irreducible V-module); (2) V is of rank 24; and (3) V_1=0; is isomorphic to the moonshine vertex operator algebra constructed by Frenkel-Lepowsky-Meurman.
In this article, we give a sufficient and necessary condition for the $C_2$-cofiniteness of the 2-cycle permutation orbifold model $(V\otimes V)^σ$ for a $C_2$-cofinite vertex operator algebra and the 2-cycle permutation $σ$ of $V\otimes V$. As an application, we show that the 2-cycle permutation orbifold model of the simple Virasoro vertex operator algebra $L(c,0)$ of minimal central charge $c$ is $C_2$-cofinite.
In this paper, we study a certain deformation $D$ of the Virasoro algebra that was introduced and called $q$-Virasoro algebra by Nigro,in the context of vertex algebras. Among the main results, we prove that for any complex number $\ell$, the category of restricted $D$-modules of level $\ell$ is canonically isomorphic to the category of quasi modules for a certain vertex algebra of affine type. We also prove that the category of restricted $D$-modules of level $\ell$ is canonically isomorphic to the category of $\mathbb{Z}$-equivariant $ϕ$-coordinated quasi modules for the same vertex algebra.…
Sub-structure in module category of Virasoro vertex operator algebras $L(c_{p,q}, 0)$
本文研究了Virasoro顶点算子代数 $L(c_{p,q},0)$ 模 category 的子结构，确定了预模和模张量子category。
We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$.Then, for any integer $t>1$, we introduce a new Lie algebra $\mathcal{L}_{t}$, and show that $σ_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$($\ell_{2}=0$)-modules are in one-to-one correspondence with restricted $\mathcal{L}_{t}$-modules of level $\ell_{13}$, where $σ_{t}$ is an order $t$ automorphism of $V_{\mathcal{L}}(\ell_{123},0)$. At the end, we give a complete list of irreducible $σ_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$($\ell_{2}=0$)-modules.
A class of vertex operator algebras generated by Virasoro vectors
本文研究一类由简单 Virasoro 向量生成的简单 OZ 型顶点算子代数，证明其由 Griess 代数唯一确定，给出了自同构群及为酉代数的必要条件。
In this note we associate to each Frobenius algebra a vertex algebra, the simplest example being the Virasoro vertex algebra. This construction is analogous to the procedure which associates to a Lie algebra with an invariant bilinear form the vacuum module over the corresponding Kac-Moody Lie algebra.
This paper consists of two parts: (1) Using a Z[1/2]-form of Virasoro vertex operator algebra L(1/2,0) with central charge 1/2, we obtain a modular vertex operator algebra over any field F of finite characteristic different from 2. We determine the generators and classify the irreducible modules for this vertex operator algebra. (2) We investigate modular framed vertex operator algebras. In particular, the rationality of modular framed vertex operator algebras is established. For a modular code vertex operator algebra, the irreducible modules are constructed and classified. Moreover, a Z[1/2]-…
Modular Zhu algebra theory and Virasoro vertex algebras
该研究利用高阶朱代数和模转换代数发展顶点代数表示理论，给出正特征域上莫比乌斯顶点代数合理性条件，证明特定离散系列维拉索罗顶点算子代数的合理性，还研究了特征$p>2$时简单商的全纯性情况。
We describe a number of relationships between properties of the vacuum Verma module of a Virasoro algebra and the automorphism group of certain vertex operator algebras. These groups include the Deligne exceptional series of simple Lie groups and some exceptional finite simple groups including the Monster and Baby Monster.
The vertex operator algebras and modules associated to the highest weight modules for the Virasoro algebra over an arbitrary field F whose characteristic is not equal to 2 are studied. The irreducible modules of vertex operator algebra L(1/2,0)_F are classified. The rationality of L(1/2,0)_F is established if the characteristic of F is different from 7.
We consider exceptional vertex operator algebras for which particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendants of the vacuum. We discuss constraints on these theories that follow from an analysis of appropriate genus zero and genus one two point correlation functions. We find explicit differential equations for the partition function in the cases where the lowest weight primary vectors form a Lie algebra or a Griess algebra. Examples include the Wess-Zumino-Witten model for Deligne's exceptional Lie algebras and the Moonshine Modu…
The work is devoted to a probably new connection between deformed Virasoro algebra and quantum $\widehat{\mathfrak{sl}}_2$. We give an explicit realization of Virasoro current via vertex operators of level 1 integrable representation of $\widehat{\mathfrak{sl}}_2$. The same is done for a twisted version of deformed Virasoro algebra.
We construct integral forms containing the conformal vector $ω$ in certain tensor powers of the Virasoro vertex operator algebra $L(\frac{1}{2},0)$, and we construct integral forms in certain modules for these algebras. When a triple of modules for a tensor power of $L(\frac{1}{2},0)$ have integral forms, we classify which intertwining operators among these modules respect the integral forms. As an application, we explore how these results might be used to obtain integral forms in framed vertex operator algebras.
In this paper we prove the fusion rules of $L(c_{1,q},0)$ for all $q\geq1$. Roughly speaking, we consider $L(c_{1,q},0)$ as the limitation of $L(c_{n,nq-1},0)$, where $n\rightarrow\infty$, and the fusion rules of $L(c_{1,q},0)$ follow as the limitation of the fusion rules of $L(c_{n,nq-1},0)$.
We study permutation orbifolds of the $2$-fold and $3$-fold tensor product for the Virasoro vertex algebra $\mathcal{V}_c$ of central charge $c$. In particular, we show that for all but finitely many central charges $\left(\mathcal{V}_c^{\otimes 3}\right)^{\mathbb{Z}_3}$ is a $W$-algebra of type $(2, 4, 5, 6^3 , 7, 8^3 , 9^3 , 10^2 )$. We also study orbifolds of their simple quotients and obtain new realizations of certain rational affine $W$-algebras associated to a principal nilpotent element. Further analysis of permutation orbifolds of the celebrated $(2,5)$-minimal vertex algebra $\mathca…
This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as are the correlation functions. We provide an example of how the Virasoro factorization algebra impleme…
We study the representation theory of the N=1 super Heisenberg-Virasoro vertex algebra at level zero, which extends the previous work on the Heisenberg-Virasoro vertex algebra arXiv:math/0201314, arXiv:1405.1707 and arXiv:1703.00531 to the super case. We calculated all characters of irreducible highest weight representations by investigating certain Fock space representations. Quite surprisingly, we found that the maximal submodules of certain Verma modules are generated by subsingular vectors. The formulas for singular and subsingular vectors are obtained using screening operators appearing i…
Primary fields of the $q$-deformed Virasoro algebra are constructed. Commutation relations among the primary fields are studied. Adjoint actions of the deformed Virasoro current on the primary fields are represented by the shift operator $Θ_ξ f(x)=f(ξx)$. Four point functions of the primary fields enjoy the connection formula associated with the Boltzmann weights of the fusion Andrews-Baxter-Forrester model.
In the references [HL1]--[HL5] and [H1], a theory of tensor products of modules for a vertex operator algebra is being developed. To use this theory, one first has to verify that the vertex operator algebra satisfies certain conditions. We show in the present paper that for any vertex operator algebra containing a vertex operator subalgebra isomorphic to a tensor product algebra of minimal Virasoro vertex operator algebras (vertex operator algebras associated to minimal models), the tensor product theory can be applied. In particular, intertwining operators for such a vertex operator algebra s…
For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge 1/2, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge 1/2 are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.
This paper studies restricted modules of gap-$p$ Virasoro algebra $Ł$ and their intrinsic connection to twisted modules of certain vertex algebras. We first establish an equivalence between the category of restricted $Ł$-modules of level $\el$ and the category of twisted modules of vertex algebra $V_{\mathcal{N}_{p}}(\el,0)$, where $\mathcal{N}_{p}$ is a new Lie algebra, $\el:=(\ell_{0},0,\cdots,0)\in\C^{\halfp+1}$, $\ell_{0}\in\C$ is the action of the Virasoro center. Then we focus on the construction and classification of simple restricted $Ł$-modules of level $\el$. More explicitly, we give…
Rationality of vertex operator superalgebras with rational conformal weights
本文证明了容许级别下的仿射顶点代数及特定仿射顶点算子超代数、极小 $W$-代数关于一族 Virasoro 元的有理性，并得出弱模范畴特定子范畴半单的结论。
Genus $g$ Virasoro Correlation Functions for Vertex Operator Algebras
本文研究简单自对偶强CFT型顶点算子代数在亏格g黎曼面上的Virasoro关联函数，证明其由微分算子作用于归一化配分函数决定，并给出该算子在Virasoro图上的显式权重表达及模性质。
For vertex operator algebra V_{\sqrt{2}A_l} associated to the even lattice \sqrt{2}A_l which is \sqrt{2} times root lattice of type A_l, it was shown by Dong-Li-Maosn-Norton that the Virasoro vector is a sum of l+1 mutually orthogonal conformal vectors with central charges c_i=1-6/(i+2)(i+3) for i=1,...,l and c_{l+1}=2l/(l+3) and the subalgebra T generated by these vectors is a tensor product of Virasoro vertex operator algebras L(c_i,0). In this paper we determine the decomposition of V_{\sqrt{2}A_3} into the sum of irreducible T-modules completely.
In this paper, a continuation of \cite{MPS}, we investigate the $S_3$-orbifold subalgebra of $(\mathcal{V}_c)^{\otimes 3}$, that is, we consider the $S_3$-fixed point vertex subalgebra of the tensor product of three copies of the universal Virasoro vertex operator algebras $\mathcal{V}_c$. Our main result is construction of a minimal, strong set of generators of this subalgebra for any generic values of $c$. More precisely, we show that this vertex algebra is of type $(2,4,6^2,8^2,9,10^2,11,12^3)$. We also investigate two prominent examples of simple $S_3$-orbifold algebras corresponding to ce…
From the defining exchange relations of the A_{q,p}(gl_{N}) elliptic quantum algebra, we construct subalgebras which can be characterized as q-deformed W_N algebras. The consistency conditions relating the parameters p,q,N and the central charge c are shown to be related to the singularity structure of the functional coefficients defining the exchange relations of specific vertex operators representations of A_{q,p}({gl_{N}) available when N=2.
Restricted modules and associated vertex algebras of extended Heisenberg-Virasoro algebra
本文引入扩展Heisenberg-Virasoro代数，构造其关联顶点代数并证明其模范畴与受限模范畴等价，进而给出简单受限模的统一构造和等价刻画。
In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $\N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For…
Let $\mathcal{F}_{25}$ be the family of irreducible highest weight modules for the Virasoro algebra of central charge $25$ which are not isomorphic to Verma modules. Let $L(25,0)$ be the Virasoro vertex operator algebra of central charge 25. We prove that the fusion rules for the $L(25,0)$-modules in $\mathcal{F}_{25}$ are in correspondence with the tensor rules for the irreducible finite dimensional representations of $sl(2, \mathbb{C})$, extending the known correspondence between modules for the Virasoro algebras of dual central charges 1 and 25.
We identify the algebra of matrix elements of big projective modules in category O with the regular functions on the big Bruhat cell of G. Analogous extensions of the regular representations of the affine Lie and Virasoro algebras yield vertex operator algebras, equipped with two commuting actions with special values of the total central charge. In the generic case, we identify the structure of these VOAs, compute their semi-infinite cohomology. Its superalgebra structure encodes the fusion rules for the associated tensor categories. Full details are provided for the affine sl(2) and Virasoro …
We establish a natural connection of the $q$-Virasoro algebra $D_{q}$ introduced by Belov and Chaltikian with affine Kac-Moody Lie algebras. More specifically, for each abelian group $S$ together with a one-to-one linear character $χ$, we define an infinite-dimensional Lie algebra $D_{S}$ which reduces to $D_{q}$ when $S=\mathbb{Z}$. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra ${\mathfrak{g}}_{S}$ with $S$ as an automorphism group and we prove that $D_{S}$ is isomorphic to the $S$-covariant algebra of the affine Lie algebra $\widehat{…
We develop a calculus of variations for functionals on certain spaces of conformal maps. Such a space Ω is composed of all maps that are conformal on domains containing a fix compact annular set of the Riemann sphere, and that are "sense-preserving". The calculus of variations is based on describing infinitesimal variations of such maps using vector fields. We show that derivatives of all orders with respect to such conformal maps, upon conjugation by an appropriate functional, give rise to the structure of the Virasoro vertex operator algebra. Our construction proceeds in three steps. We firs…
We consider a class of weak modules for vertex operator algebras that we call logarithmic modules. We also construct nontrivial examples of intertwining operators between certain logarithmic modules for the Virasoro vertex operator algebra. At the end we speculate about some possible logarithmic intertwiners at the level $c=0$.
Conformal blocks, parenthesized braid operad, and $c=1/2$ Virasoro vertex operator algebra
本文通过共形块和路径解析延拓构造顶点算子代数模范畴的伪辫子结构，计算中心荷 $1/2$ 的 Virasoro 顶点算子代数的四点共形块，并证明其模范畴等价于 $\mathbb{Z}_2$ 上的 Tambara-Yamagami 范畴。
In this paper, we study certain Virasoro frames for lattice vertex operator algebras and their $\Z_2$-orbifolds using linear codes over $\Z_4$. We also compute the corresponding frame stabilizer from the view point of binary codes and $\Z_4$-codes. As an application, we determine the frame stabilizers of several Virasoro frames of the vertex operator algebra $V_{E_8}$ and the moonshine vertex operator algebra $V^\natural$.
In this paper we discuss the structure of the tensor product V'_{α,β}\otimes L(c,h) of irreducible module from intermediate series and irreducible highest weight module over the Virasoro algebra. We generalize Zhang's irreducibility criterion, and show that irreducibility depends on the existence of integral roots of a certain polynomial, induced by a singular vector in the Verma module V(c,h). A new type of irreducible Vir-module with infinite-dimensional weight subspaces is found. We show how the existence of intertwining operator for modules over vertex operator algebra yields reducibility …
An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra $U_q(\hat{sl}_2)$. A similar construction is proposed for the elliptic algebra $A_{q,p}(\hat{sl}_2)$.
We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the bilinear trace procedure proposed in the 90's. It allows us to make contact with the vertex operator techniques that were introduced separately at the same period. As a by-product, the method pinpoints two critical values of the central charge for which the center of the algebra is extended, as well as (in the $gl(2)$ case) a Liouville formula.
The rational and C_2-cofinite simple vertex operator algebras whose effective central charges and the central charges c are equal and less than 1 are classified. Such a vertex operator algebra is zero if c<0 and C if c=0. If c>0, it is an extension of discrete Virasoro vertex operator algebra L(c_{p,q},0) by its irreducible modules. It is also proved that for any rational and C_2-cofinite simple vertex operator algebra whose effective central charge and central charge are equal, the vertex operator subalgebra generated by the Virasoro vector is simple.
Three examples of free field constructions for the vertex operators of the elliptic quantum group ${\cal A}_{q,p}(\hat{sl}_2)$ are obtained. Two of these (for $p^{1/2}=\pm q^{3/2},p^{1/2}=-q^2$) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 $Z$-algebra of Lepowsky and Wilson. The third one ($p^{1/2}=q^{3}$) is constructed over a tensor product of a bosonic and a fermionic Fock spaces. The algebraic structure at $p^{1/2}=q^{3}$, however, is not related to the deformed Virasoro algebra. Using these free field constructions, an …
As the Yangian double with center,which is deformed from affine algebra by the additive loop parameter $\hbar$ ,we get the commuting relation and the bosonization of quantum $\hbar$-deformed Virasoro algebra. The corresponding Miura transformation, associated screening operators and the BRST charge have been studied. Moreover, we also constructe the bosonization for type I and type II intertwiner vertex operators. Finally, we show that the commuting relation of these vertex operators in the case of $p'=r p=r-1$ and $\hbar =π$ actually gives the exact scattering matrix of the Restricted sine-Go…
Virasoro frames and their Stabilizers for the E_8 lattice type Vertex Operator Algebra
本文研究 E_8 格型顶点算子代数的 Virasoro 框架稳定化子，证明其自同构群作用恰有五个轨道，并给出稳定化子群结构的范畴化描述。
We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of $\mathfrak{osp}(1|2)$ and the $N=1$ super Virasoro algebra to categories of Virasoro algebra modules via certain …
We determine the decomposition of V_{\sqrt{2}D_l} into a sum of irreducible T-modules for general l where D_l is the root lattice of type D_l and T is the tensor product of l+1 Virasoro vertex operator algebras with central charges c_{1}=1/2, c_{2}=7/10, c_{3}=4/5, and c_{i}=1 for 4\le i\le l+1.
Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Vir…
Conformal nets from minimal W-algebras
本文证明了所有幺正极小W-代数的强分级局部性，从而产生不可约分级局部共形网，并基于此提出了统一构造以获得包含N=3,4超Virasoro网在内的新共形网，同时证明了完全有理猜想对N=0至4超Virasoro代数成立。
In this paper, we study contragredient duals and invariant bilinear forms for modular vertex algebras (in characteristic $p$). We first introduce a bialgebra $\mathcal{H}$ and we then introduce a notion of $\mathcal{H}$-module vertex algebra and a notion of $(V,\mathcal{H})$-module for an $\mathcal{H}$-module vertex algebra $V$. Then we give a modular version of Frenkel-Huang-Lepowsky's theory and study invariant bilinear forms on an $\mathcal{H}$-module vertex algebra. As the main results, we obtain an explicit description of the space of invariant bilinear forms on a general $\mathcal{H}$-mo…
The admissible modules for $\hat{sl}_2$ are studied from the point of view of vertex operator algebra. If $l$ is rational such that $l+2={p\over q}$ for some coprime positive integers $p\ge 2$ and $q$, Kac and Wakimoto found finitely many distinguished irreducible representations for $\hat{sl}_2$, called admissible representations. In this paper we prove that the vertex operator algebra $L(l,0)$ associated to irreducible highest weight representation of $l$ is not rational if $l$ is not a positive integer. However if we change the Virasoro algebra in certain way, $L(l,0)$ becomes a rational ve…
In this paper, we give a characterization of the rational vertex operator algebra VTL, where L is the root lattice of type A1 and T is the tetrahedral group.
This is a paper in a series to study quantum vertex algebras and their relations with various quantum algebras. In this paper, we introduce a notion of T-type quantum vertex algebra and a notion of $G$-covariant $ϕ$-coordinated quasi module for a $T$-type quantum vertex algebra with an automorphism group $G$. We refine and extend several previous results and we obtain a commutator formula for $G$-covariant $ϕ$-coordinated quasi modules. As an illustrating example, we study a special case of the deformed Virasoro algebra $\V_{p,q}$ with $q=-1$, to which we associate a Clifford vertex superalgeb…
On Virasoro-type reductions and inverse Hamiltonian reductions for $W$-algebras and $W_\infty$-algebras
本文为经典李型及高度二幂零轨道相关的 $W$-代数建立 Virasoro 型约化与逆约化，并推广至 $W_\infty$-代数等泛对象。
In this paper, we present a general construction of 3-transposition groups as automorphism groups of vertex operator algebras. Applying to the moonshine vertex operator algebra, we establish the Conway-Miyamoto correspondences between Fischer 3-transposition groups $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ and $c=25/28$ and $c=11/12$ Virasoro vectors of subalgebras of the moonshine vertex operator algebra.
It is proved that any vertex operator algebra for which the image of the Virasoro element in Zhu's algebra is algebraic over complex numbers is finitely generated. In particular, any vertex operator algebra with a finite dimensional Zhu's algebra is finitely generated. As a result, any rational vertex operator algebra is finitely generated.
We show that there is a braided tensor category structure on the category of $C_1$-cofinite modules for the (universal or simple) Virasoro vertex operator algebras of arbitrary central charge. In the generic case of central charge $c=13-6(t+t^{-1})$, with $t \notin \mathbb{Q}$, we prove semisimplicity, rigidity and non-degeneracy and also compute the fusion rules of this tensor category.
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