Log-Modular Quantum Groups at Even Roots of Unity and the Quantum Frobenius I

Abstract

We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been predicted by certain conformal field theory considerations, but constructions had not appeared until recently. We show that our quantum groups can be identified with those of Creutzig-Gainutdinov-Runkel in type $A_1$, and Gainutdinov-Lentner-Ohrmann in arbitrary Dynkin type. We discuss conjectural relations with vertex operator algebras at (1, p)-central charge. For example, we explain how one can (conjecturally) employ known linear equivalences between the triplet vertex algebra and quantum $\mathfrak {sl}_2$, in conjunction with a natural ${{\,\mathrm{PSL}\,}}_2$-action on quantum $\mathfrak {sl}_2$ provided by our de-equivariantization construction, in order to deduce linear equivalences between “extended” quantum groups, the singlet vertex operator algebra, and the (1, p)-Virasoro logarithmic minimal model. We assume some restrictions on the order of our root of unity outside of type $A_1$, which we intend to eliminate in a subsequent paper.

Appendix A: Details on Rational (De-)equivariantization

We cover the details needed to prove Proposition 8.6. As a first order of business let us provide the proof of Lemma 8.4.

Proof of Lemma 8.4

The fact that any finitely presented object is compact follows from the fact that free objects $unit_*V$, for V in $\mathscr {D}$, are compact, and left exactness of the ${{\,\mathrm{Hom}\,}}$ functor. Now, for arbitrary M in $\mathscr {D}_S$ we may write M as the union $M=\varinjlim _\alpha M'_\alpha$ of its finitely generated submodules $M'_\alpha$. For any finitely generated $M'$ we may write the kernel N of a projection $unit_*V'=S\otimes _\mathbb {C} V'\rightarrow M'$ as a direct limit of finitely generated modules $N=\varinjlim _\beta N_\beta$ and hence write $M'$ as a direct limit of finitely presented modules $M'=\varinjlim _\beta M_\beta$, with $M_\beta =S\otimes _\mathbb {C} V'/N_\beta$. Thus we may write arbitrary M as a direct limit $M=\varinjlim _\kappa M_\kappa$ of finitely presented modules. Compactness of M implies that the identity factors through some finitely presented $M_\kappa$, and hence $M=M_\kappa$.

1.1 Equivariantization and the de-equivariantization

Suppose $F:{{\,\mathrm{rep}\,}}\Pi \rightarrow \mathscr {C}$ is a central embedding which is faithfully flat and locally finite. Take

$$\begin{aligned} R:=\mathscr {O}\text { considered as a algebra object in }{{\,\mathrm{rep}\,}}\Pi \text { with trivial } \Pi -\text {action}. \end{aligned}$$

We omit the prefix F and write simply write $\mathscr {O}$ and R for the images of these algebras in $\mathscr {C}$. We define the functor on the de-equivariantization

$$\begin{aligned} \psi _u:\mathscr {C}_\Pi \rightarrow (\mathscr {C}_\Pi )_R,\ \ \psi _u M:=R\otimes M, \end{aligned}$$

where $\mathscr {O}$ acts diagonally on each $\psi _uM$ and R acts via the first component. More precisely, we have the algebra map $\Delta :\mathscr {O}\rightarrow R\otimes \mathscr {O}$ in ${{\,\mathrm{rep}\,}}\Pi$ given by comultiplication and act naturally on $\psi _uM$ via $\Delta$. For finite presentation, one observes on free modules $\mathscr {O}\otimes V$ an easy isomorphism $\psi _u(\mathscr {O}\otimes V)\cong unit_*(\mathscr {O}\otimes V)$ in $(\mathscr {C}_\Pi )_R$, so that applying $\psi _u$ to a finite presentation for M, as an $\mathscr {O}$-module, yields a finite presentation for $\psi _uM$ over R.

We have the natural iosmorphism

$$\begin{aligned} \psi _u\psi _u(V)=R\otimes (R\otimes V)\cong (R\otimes R)\otimes V=\Delta _*\psi _u(V) \end{aligned}$$

given by the associativity in $\mathscr {C}$ and the natural isomorphism $\psi _u V\otimes _{(R\otimes \mathscr {O})} \psi _u W\cong \psi _u(V\otimes _\mathscr {O}W)$ given by multiplication from R. Whence we have a canonical rational action of $\Pi$ on the de-equivariantization $\mathscr {C}_\Pi$, and can consider the corresponding equivariantization $(\mathscr {C}_\Pi )^\Pi$. Objects in this category are simply $\mathscr {O}$-modules in $\mathscr {C}$ with a compatible R-coaction.

Note that the R-coinvariants $X^R$ of an equivariant object X is a $\mathscr {C}$-subobject in X, as it is the preimage of $\mathbf {1}\otimes X\subset R\otimes X$ under the R-coaction. Whence we have the functor

$$\begin{aligned} (-)^R:(\mathscr {C}_\Pi )^\Pi \rightarrow {{\,\mathrm{Ind}\,}}\mathscr {C},\ \ X\mapsto X^R. \end{aligned}$$

In addition, for any V in $\mathscr {C}$ the object ${{\,\mathrm{can}\,}}^!(V)=\mathscr {O}\otimes V$ can be given the $\mathscr {O}$-action and R-coaction from $\mathscr {O}$. The coinvariants of ${{\,\mathrm{can}\,}}^!(V)$ is the subobject $\mathbf {1}\otimes V$, and the unital structure on $\mathscr {C}$ provides a natural ismorphism $\zeta :(-)^R\circ {{\,\mathrm{can}\,}}^!\overset{\sim }{\rightarrow }id_\mathscr {C}$. We also have the natural transformation $\gamma :{{\,\mathrm{can}\,}}^!\circ (-)^R\rightarrow id_{(\mathscr {C}_\Pi )^\Pi }$ given by the $\mathscr {O}$-action

$$\begin{aligned} \gamma _X:{{\,\mathrm{can}\,}}^!(X^R)=\mathscr {O}\otimes X^R\rightarrow X. \end{aligned}$$

Lemma A.1

The transformation $\gamma$ is a natural isomorphism, and the coinvariants functor $(-)^R$ has image in $\mathscr {C}$.

Proof

We have the twisted comultiplication $\Delta ^S:R\rightarrow \mathscr {O}\otimes \mathscr {O}$, $f\mapsto f_1\otimes S(f_2)$, and can define the inverse $\gamma _X^{-1}:X\rightarrow \mathscr {O}\otimes X^R$ as the composite

$$\begin{aligned} X\overset{\rho }{\rightarrow }R\otimes X\overset{\Delta ^S\otimes 1}{\rightarrow }\mathscr {O}\otimes \mathscr {O}\otimes X\rightarrow \mathscr {O}\otimes X, \end{aligned}$$

which one can check has image in $\mathscr {O}\otimes X^R$ and does in fact provide the inverse to $\gamma$, just as in the Hopf case [54]. To see that $X^R$ is in $\mathscr {C}$, and not in ${{\,\mathrm{Ind}\,}}\mathscr {C}{\setminus }\mathscr {C}$, we note that $X\cong \mathscr {O}\otimes X^R$ is of finite length in $\mathscr {C}_\Pi$ and that $\mathscr {O}\otimes -$ is exact, which forces $X^R$ to be of finite length. Hence $X^R$ is in $\mathscr {C}$.

Since both $\zeta$ and $\gamma$ are isomorphisms we have directly

Proposition A.2

(cf. [11, 22]) The functor ${{\,\mathrm{can}\,}}^!:\mathscr {C}\rightarrow (\mathscr {C}_\Pi )^\Pi$ is an equivalence of monoidal (and hence tensor) categories.

Remark A.3

One can avoid all finiteness concerns by employing the ${{\,\mathrm{Ind}\,}}$-category ${{\,\mathrm{Ind}\,}}\mathscr {C}$ and the category of arbitrary modules $\mathscr {O}\text {-Mod}_{{{\,\mathrm{Ind}\,}}{\mathscr {C}}}$. Then, with the cocomplete theory of Sect. 8.1, one can argue exactly as above to find that the functor ${{\,\mathrm{can}\,}}^!:{{\,\mathrm{Ind}\,}}\mathscr {C}\rightarrow (\mathscr {O}\text {-Mod}_{{{\,\mathrm{Ind}\,}}\mathscr {C}})^\Pi$ is again an equivalence.

1.2 De-equivariantizing the equivariantization

Let $\mathscr {D}$ be a tensor category equipped with a rational action of $\Pi$. There is a canonical embedding ${{\,\mathrm{rep}\,}}\Pi \rightarrow \mathscr {D}^\Pi$ into the equivariantization which identifies ${{\,\mathrm{rep}\,}}\Pi$ with the preimage of $Vect\subset \mathscr {D}$ in $\mathscr {D}^\Pi$, under the forgetful functor. Indeed, the fact that the action map $\psi _u:\mathscr {D}\rightarrow \mathscr {D}_R$ is monoidal implies that $\psi _u(\mathbf {1})=R$, so that the restriction of $\psi _u$ to the trivial subcategory $Vect\subset \mathscr {D}$ is equated with the usual action of $\Pi$ on Vect, and hence $Vect^\Pi ={{\,\mathrm{rep}\,}}\Pi$.

We have the two algebras $\mathscr {O}$ and R in ${{\,\mathrm{rep}\,}}\Pi$, the latter one being trivial, which are equated under the composite ${{\,\mathrm{rep}\,}}\Pi \rightarrow \mathscr {D}^\Pi \rightarrow \mathscr {D}$, i.e. which are indistinguishable as objects in $\mathscr {D}$. Hence the counit $\mathscr {O}\rightarrow \mathbf {1}$, which is not a map in ${{\,\mathrm{rep}\,}}\Pi$, is a map in $\mathscr {D}$, and for any $\mathscr {O}$-module in the equivariantization $\mathscr {D}^\Pi$ the reduction $X_\mathscr {O}:=\mathbf {1}\otimes _\mathscr {O}X$ is a well-defined object in $\mathscr {D}$.

Since $\mathscr {O}$ is trivial in $\mathscr {D}$, and $\psi _u$ is a tensor map, we have $\psi _u(\mathscr {O})=R\otimes \mathscr {O}$. By the definition of $\mathscr {O}$ in ${{\,\mathrm{rep}\,}}\Pi$ the equivariant structure is given by the comultiplication $\Delta :\mathscr {O}\rightarrow R\otimes \mathscr {O}$. Hence $\mathscr {O}$ acts naturally on each $\psi _u(X)$ via the comultiplication, for any $\mathscr {O}=R$-module X in $\mathscr {D}$. So we can consider $\mathscr {O}$-modules in $\mathscr {D}^\Pi$ as $\mathscr {O}=R$-modules in $\mathscr {D}$ for which the coaction $X\rightarrow \psi _u(X)$ is $\mathscr {O}$-linear.

For any object V in $\mathscr {D}$ we consider V as a trivial $\mathscr {O}$-module, and let $\mathscr {O}$ act on $\psi _u(V)$ diagonally. Each $\psi _u(V)$ then becomes an object in $(\mathscr {D}^\Pi )_\Pi$ via the “free” coaction, $\psi _u(V)\rightarrow \psi _u\psi _u(V)$ given by the unit of the $(\Delta _*,\Delta ^*)$-adjunction

$$\begin{aligned} \psi _u\overset{unit}{\rightarrow }\Delta ^*\Delta _*\psi _u\overset{\Delta ^*\sigma }{\rightarrow }\psi _u\psi _u. \end{aligned}$$

We have the reduction functor $1^*:(\mathscr {D}^\Pi )_\Pi \rightarrow \mathscr {D}$, $X\mapsto X_\mathscr {O}$, and the free functor ${{\,\mathrm{can}\,}}_!:\mathscr {D}\rightarrow (\mathscr {D}^\Pi )_\Pi$, $V\mapsto \psi _u(V)$. There are natural transformations

$$\begin{aligned} \eta _V:\psi _u(V)_\mathscr {O}=1^*\psi _u(V)\overset{\sim }{\rightarrow }V,\ \ \eta :1^*\circ {{\,\mathrm{can}\,}}_!\overset{\sim }{\rightarrow }id_{\mathscr {D}}, \end{aligned}$$

and

$$\begin{aligned} \vartheta _X:X\rightarrow \psi _u(X_\mathscr {O}),\ \ \vartheta :id_{(\mathscr {D}^\Pi )_\Pi }\rightarrow {{\,\mathrm{can}\,}}_!\circ 1^*, \end{aligned}$$

the former of which is simply given by the counit for $\psi _u$ and the latter is given as the composite $X\rightarrow \psi _u(X)\rightarrow \psi _u(X_\mathscr {O})$ of the comultiplication and the application of $\psi _u$ to the reduction $X\rightarrow X_\mathscr {O}$ in $\mathscr {D}$. The following is a consequence of the fact that each object in $(\mathscr {D}^\Pi )_\Pi$ is finitely presented over $\mathscr {O}$.

Lemma A.4

The transformation $\vartheta$ is a natural isomorphism if and only if it is a natural isomorphism when applied to free modules $\mathscr {O}\otimes W$, for W in $\mathscr {D}^\Pi$.

Lemma A.5

An object X is 0 in $(\mathscr {D}^\Pi )_\Pi$ if and only if the fiber $1^*X$ is 0.

Proof

We may write $\mathscr {D}={{\,\mathrm{corep}\,}}C$ for a coalgebra C, by Takeuchi reconstruction [64]. Then $\mathscr {D}_R$ is just the category of corepresentations of the R-coalgebra $C_R$ which are finitely presented over R. Now, for a finitely presented R-module M we understand that M vanishes if and only if its fiber $x^*M$ vanishes for each closed point $x:{{\,\mathrm{Spec}\,}}(K)\rightarrow \Pi$. Let $p(x):\mathscr {O}_K\rightarrow K$ be the corresponding ring map. Note that the reduction simply takes the fiber at the identity.

Take M in $(\mathscr {D}^\Pi )_\Pi$ and suppose that $1^*M$ vanishes. Consider a closed point $x\in \Pi (K)$. By changing base to $\mathscr {D}_K$ and $\Pi _K$ we may assume that K is our base field, so that $x^{-1}\cdot x=\epsilon$. Via the the coaction we find an isomorphism

$$\begin{aligned} M\overset{\rho _M}{\rightarrow }\psi _u M\rightarrow p(x)_*\psi _u M=t_x M, \end{aligned}$$

(8)

where the last map is the counit of the $(p(x)_*,p(x)^*)$-adjunction, and $t:\Pi (K)\rightarrow {{\,\mathrm{Aut}\,}}(\mathscr {D})$ is the discrete action of $\Pi (K)$.

Now, $t_x M$ has a canonical $\mathscr {O}=R$-action via the functorial identification ${{\,\mathrm{End}\,}}_\mathscr {D}(M)\cong {{\,\mathrm{End}\,}}_\mathscr {D}(t_xM)$, and the fiber $y^*M$ at a given K-point y vanishes if and only if the fiber $y^*(t_x M)$ vanishes. If we let $f_x:R\rightarrow R$ denote the automorphism given by left translation by x then we see that (8) is an R-linear isomorphism from M to the restriction of $t_xM$ along $f_x$. In particular, we have

$$\begin{aligned} 0=1^*M\cong 1^*({\text {res}}_{f_x}t_xM)=x^*(t_x M), \end{aligned}$$

which implies $x^*M=0$. Since x was arbitrary, we see $M=0$ if $1^*M=0$. Conversely, the fiber at the identity obviously vanishes if M vanishes.

Proposition A.6

The functor ${{\,\mathrm{can}\,}}_!:\mathscr {D}\rightarrow (\mathscr {D}^\Pi )_\Pi$ is an equivalence of monoidal (and hence tensor) categories. Furthermore, the embedding $F:{{\,\mathrm{rep}\,}}\Pi \rightarrow \mathscr {D}^\Pi$ is faithfully flat and locally finite

Proof

We prove that $\vartheta$ is an isomorphism on free modules. Take $T=\mathscr {O}\otimes V$ consider $\vartheta _T:T\rightarrow \psi _u(V)$. We extend to a right exact sequence $T\rightarrow \psi _u(V)\rightarrow M\rightarrow 0$. The counital property for $\psi _u$ implies that the fiber $1^*\vartheta$ is identified with the identity on V. By right exactness of the reduction we have $1^*M=0$, and hence the cokernel vanishes by Lemma A.5.

We now extent $\vartheta _T$ to a left exact sequence $T'\overset{p}{\rightarrow }T\overset{\vartheta _T}{\rightarrow }\psi _u(V)\rightarrow 0$, with p a map from a finite free module. (We need to use the fact that $\psi _u(V)$ is finitely presented to verify that such an extension exists). Since $\psi _u$ is a monoidal functor it preserves duals [24, Exercise 2.10.6], it follows that $\psi _u(V)$ is dualizable in $\mathscr {D}_R$ with dual $\psi _u(V)^\vee \cong \psi _u(V^*)$. Free modules $R\otimes W$ are also dualizable with dual $R\otimes W^*$.

Note that $1^*:(\mathscr {D}_\Pi )^\Pi \rightarrow \mathscr {D}$ is a monoidal functor, and hence preserves duality as well, so that $1^*(\vartheta _T^\vee )$ is identified with the isomorphism $(1^*\vartheta _T)^*$. So by the same arguments employed above the dual $\vartheta _T^\vee :\psi _u(V)^\vee \rightarrow T^\vee$ is also surjective. Since the dual composite

$$\begin{aligned} \psi _u(V)^\vee \rightarrow T^\vee \overset{p^\vee }{\rightarrow }(T')^\vee \end{aligned}$$

is 0 we find that $p^\vee$ is 0. Since duality $(-)^\vee$ is an equivalence on the category of (left and right) dualizable objects in $(\mathscr {D}^\Pi )_\Pi$, it follows that $p=0$. So $\vartheta _T$ is an isomorphism for each free T. We now employ Lemma A.4 to find that ${{\,\mathrm{can}\,}}_!$ is an equivalence. The fact that $\mathscr {D}$ is a tensor category and that ${{\,\mathrm{can}\,}}_!$ is an equivalence implies that F is both faithfully flat and locally finite.

1.3 Proof of Proposition 8.6

Proof of Proposition 8.6

Take $\mathscr {D}=\mathscr {C}_\Pi$. We have the de-equivariantization functor $\mathscr {C}\rightarrow \mathscr {D}$. For a sequence ${{\,\mathrm{rep}\,}}\Pi \rightarrow \mathscr {K}\overset{i}{\rightarrow }\mathscr {K}'\rightarrow \mathscr {C}$ we have the de-equivariantization $\mathscr {K}_\Pi \overset{i_\Pi }{\rightarrow }\mathscr {K}'_\Pi \rightarrow \mathscr {D}$, with $\mathscr {K}_\Pi$ and $\mathscr {K}'_\Pi$ stable under the action of $\Pi$. By the definition of the equivalence of ${{\,\mathrm{can}\,}}^!$, in Sect. A.1, we find that there is a diagram

Hence i is an equivalence if and only if $i_\Pi$ is an equivalence, and thus de-equivariantization $(-)_\Pi$ defines an inclusion of the poset of (isomorphism-closed) intermediate categories $\Pi \text {-Int}(\mathscr {C})=\{{{\,\mathrm{rep}\,}}\Pi \subset \mathscr {K}\subset \mathscr {C}\}$ to the poset $\Pi \text {-Stab}(\mathscr {D})=\{\mathscr {W}\subset \mathscr {D}\}$ of (isomorphism-closed) $\Pi$-stable categories. A completely similar argument, using ${{\,\mathrm{can}\,}}_!$, shows that equivariantization $\mathscr {W}\subset \mathscr {D}\rightsquigarrow \mathscr {W}^\Pi \subset \mathscr {C}$ defines an inclusion of posets $\Pi \text {-Stab}(\mathscr {D})\rightarrow \Pi \text {-Int}(\mathscr {C})$ which is inverse to $(-)_\Pi$.

Since de-/equivariantization under a central inclusion/braided action preserves braided subcategories, and central subcategories, the above argument shows that this bijection of posets restricts to a bijection for both braided and central subcategories as well.