Tensor categories of affine Lie algebras beyond admissible levels

Abstract

We show that if V is a vertex operator algebra such that all the irreducible ordinary V-modules are $C_1$-cofinite and all the grading-restricted generalized Verma modules for V are of finite length, then the category of finite length generalized V-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) $\mathfrak {g}$ at level k and the category $KL_k(\mathfrak {g})$ of its finite length generalized modules, we discover several families of $KL_k(\mathfrak {g})$ at non-admissible levels k, having braided tensor category structures. In particular, $KL_k(\mathfrak {g})$ has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories $KL_k(\mathfrak {g})$, including the category $KL_{-1}(\mathfrak {sl}_n)$. Using these results, we construct a rigid tensor category structure on a full subcategory of $KL_1(\mathfrak {sl}(n|m))$ consisting of objects with semisimple Cartan subalgebra actions.

Appendix

1.1 Change of basis for the root system of $\mathfrak {sl}(n|m)$

For the Lie superalgebra $\mathfrak {sl}(n|m)$, there is no unique simple root system and we choose a distinguished simple root system, i.e. we will only have one odd positive simple root. This is easiest described by embedding it into $\mathbb {Z}^{n+m} = \mathbb {Z} \epsilon _1 \oplus \dots \oplus \mathbb {Z} \epsilon _n \oplus \mathbb {Z} \delta _1 \dots \oplus \mathbb {Z}\delta _m$ with norm $\epsilon _i\epsilon _j = \delta _{i, j}$ and $\delta _i\delta _j = -\delta _{i, j}$ and elements $a_i \epsilon _1 + \dots a_n \epsilon _n + b_1\delta _1 + \dots b_m \delta _m$ of the root lattice satisfy $a_1+ \dots + a_n +b_1 + \dots + b_m=0$. For example, we denote $\Delta _{\overline{0}}^+ \cup \Delta _{\overline{1}}^+$ as a system of positive roots, and denote by $\Pi$ the corresponding set of positive roots and our choice is

$$\begin{aligned} \begin{aligned} \Delta _{\overline{0}}^+&= \{\epsilon _i-\epsilon _j, \delta _i-\delta _j|i < j\}, \qquad \qquad \Delta _{\overline{1}}^+ = \{\epsilon _i-\delta _j\},\\ \Pi&= \{ \alpha _1, \dots , \alpha _{n+m-1}\}, \\ \alpha _i :&= {\left\{ \begin{array}{ll} \epsilon _i-\epsilon _{i+1}, &{} \quad i=1, \dots , n-1 \\ \epsilon _n-\delta _1, &{} \quad i=n \\ \delta _{m+n-i}-\delta _{m+n+1-i}, &{} \quad i=n+1, \dots , m+n-1 \end{array}\right. }, \end{aligned} \end{aligned}$$

(7.1)

see for example [75] for a reference for the Lie superalgebra $\mathfrak {sl}(n|m)$ and its root system.

Let us set $\epsilon := \epsilon _1 + \dots + \epsilon _n$ and $\delta := \delta _1+ \dots + \delta _m$. The corresponding fundamental weights are

$$\begin{aligned} \begin{aligned} \omega _i&= \frac{1}{n-m} \left( (n-m-i) (\epsilon _1 +\cdots \epsilon _i) - i(\epsilon _{i+1} + \cdots + \epsilon _n -\delta _1 - \cdots - \delta _m) \right) \\&= \frac{i}{n-m}(\delta -\epsilon ) + \epsilon _1 + \dots + \epsilon _i. \end{aligned} \end{aligned}$$

for $i = 1, \dots , n-1$,

$$\begin{aligned} \omega _n= & {} \frac{1}{n-m}(-m(\epsilon _1 + \cdots + \epsilon _{n}) + n(\delta _1 +\cdots + \delta _m)) \\= & {} \frac{-m\epsilon +n\delta }{n-m} = \frac{n}{n-m}(\delta -\epsilon ) + \epsilon \end{aligned}$$

and

$$\begin{aligned} \omega _{n+j} = \frac{j}{n-m}(\delta -\epsilon ) + \delta _{m-j+1} + \dots + \delta _m \end{aligned}$$

(7.2)

for $j = 1, \dots , m-1$. The positive roots of the even subalgebra are expressed in terms of ours as

$$\begin{aligned} \{ \alpha _1, \dots , \alpha _{n-1}, \mu \omega _n, \alpha _{n+1}, \dots , \alpha _{n+m-1} \}, \end{aligned}$$

where $\mu = \sqrt{\frac{m-n}{mn}}$ as in Sect. 6.1 and the corresponding fundamental weights are

$$\begin{aligned} \begin{aligned} \nu _i&= -\frac{i}{n}\epsilon + \epsilon _1 + \dots \epsilon _i = \omega _i - \frac{i}{n} \omega _n\\ \nu _n&= -\mu \omega _n \\ \nu _{n+j}&= \frac{j}{m}\delta - (\delta _1 + \dots \delta _j) = \omega _{n+m-j} -\frac{j}{m} \omega _n. \end{aligned} \end{aligned}$$

(7.3)

Thus weight labels translate as

$$\begin{aligned} \begin{aligned} \Lambda&= \sum _{i = 1}^{n+m-1}a_i\nu _i = \sum _{i = 1}^{n+m-1}b_i\omega _i \qquad \text {with} \\ b_i&= a_i, \qquad b_n = -\mu a_n -\frac{1}{n} \sum _{i=1}^{n-1} ia_i - \frac{1}{m} \sum _{j=1}^{m-1} ja_j\qquad \text {and} \qquad b_{n+j} = a_{n+j} \end{aligned} \end{aligned}$$

for $i=1, \dots , n-1$ and $j=1, \dots , m-1$.

1.2 Top level of the induced module

Lemma 7.1

The conformal gradings of the induced modules $\mathcal {F}(M_{a, s,\lambda })$ are bounded from below and the bound is given by

$$\begin{aligned} \frac{n}{2}(z+1)^2 - n\left( \frac{a}{n}+\frac{1}{2}+z\right) (z+1) + \frac{a(n-a)}{2n} + \frac{s(s+m)}{2m} + \frac{\lambda ^2}{2} \end{aligned}$$

if $-zn-a+s$ is nonnegative; the bound is

$$\begin{aligned} \frac{n}{2}z^2 - zn\left( \frac{a}{n}-\frac{1}{2}+z\right) + \frac{a(n-a)}{2n} + \frac{s(s-m)}{2m} + \frac{\lambda ^2}{2} \end{aligned}$$

if $-zn-a+s$ is nonpositive.

The proof will be done by analyzing two cases. Let $t = rn + \bar{t}$ for $0\le \bar{t} < n$ and $r \in \mathbb {Z}$.

Case 1 If $s+t \ge 0$,

$$\begin{aligned}&h_{s+t, \overline{a+t}, \lambda +\mu t} \\&\quad = \frac{(\overline{a+t})(n-\overline{a+t})}{2n} + \frac{(s+t)(s+t+m)}{2m} + \frac{(\lambda + \mu t)^2}{2}\\&\quad = \frac{(\overline{a+t})(n-\overline{a+t})}{2n} + \frac{(s+rn+\overline{t})(s+rn+\overline{t}+m)}{2m} + \frac{(\lambda + \mu (rn+\overline{t}))^2}{2}\\&\quad = \bigg (\frac{\mu ^2n^2}{2}+\frac{n^2}{2m}\bigg )r^2 + \bigg (n\big (\mu ^2+\frac{1}{m}\big )\overline{t}+\frac{2ns+mn}{2m}+\lambda \mu n\bigg )r + \text{ const }. \end{aligned}$$

Since $\mu ^2+\frac{1}{m} = \frac{1}{n}$ is always positive, the conformal weights $\{h_{s+t, \overline{a+t}, \lambda +\mu t}|t \in \mathbb {Z}\}$ always have a lower bound. The minimal values attain when r is an integer that is closest to

$$\begin{aligned} -\frac{\overline{t}}{n} - \frac{s}{m} - \frac{1}{2} - \lambda \mu = -z-\frac{\overline{t}+a}{n} - \frac{1}{2}. \end{aligned}$$

We list the minimal values for all $\overline{t}$ as follows:

• If $\overline{t} + a = 0$, i.e., $a = 0, \overline{t} = 0$, then $r = -z$ or $-z-1$.

• If $1 \le \overline{t} + a \le n-1$, then $r = -z-1$.

• If $\overline{t} + a = n$, then $r = -z-1$ or $-z-2$.

• If $\overline{t} + a \ge n+1$, then $r = -z-2$.

The minimum are all the same (independent of $\overline{t}$):

$$\begin{aligned} \frac{n}{2}(z+1)^2 - n\left( \frac{a}{n}+\frac{1}{2}+z\right) (z+1) + \frac{a(n-a)}{2n} + \frac{s(s+m)}{2m} + \frac{\lambda ^2}{2} \end{aligned}$$

(7.4)

It is easy to see (7.4) attains when $-zn-a+s$ is nonnegative, i.e. $\mu ^2 s -\lambda \mu \ge 0$.

Case 2 If $s+t < 0$,

$$\begin{aligned}&h_{s+t,\overline{a+t}, \lambda +\mu t} \\&\quad = \frac{(\overline{a+t})(n-\overline{a+t})}{2n} + \frac{(s+t)(s+t-m)}{2m} + \frac{(\lambda + \mu t)^2}{2}\\&\quad = \bigg (\frac{\mu ^2n^2}{2}+\frac{n^2}{2m}\bigg )r^2 + \bigg (n\big (\mu ^2+\frac{1}{m}\big )\overline{t}+\frac{2ns-mn}{2m}+\lambda \mu n\bigg )r + \text{ const }. \end{aligned}$$

The minimal values attain when r is an integer that is closest to

$$\begin{aligned} -\frac{\overline{t}}{n} - \frac{s}{m} + \frac{1}{2} - \lambda \mu = -z-\frac{\overline{t}+a}{n} + \frac{1}{2}. \end{aligned}$$

We list the minimal values for all $\overline{t}$ as follows:

• If $\overline{t} + a = 0$, i.e., $a = 0, \overline{t} = 0$, then $r = -z$ or $-z+1$.

• If $1 \le \overline{t} + a \le n-1$, then $r = -z$.

• If $\overline{t} + a = n$, then $r = -z-1$ or $-z$.

• If $\overline{t} + a \ge n+1$, then $r = -z-1$.

The minimum are all the same (independent of $\overline{t}$):

$$\begin{aligned} \frac{n}{2}z^2 - zn\left( \frac{a}{n}-\frac{1}{2}+z\right) + \frac{a(n-a)}{2n} + \frac{s(s-m)}{2m} + \frac{\lambda ^2}{2}. \end{aligned}$$

(7.5)

Similar to the previous case, (7.5) attains when $-zn-a+s$ is nonpositive.