Positive line modules over the irreducible quantum flag manifolds

Abstract

Noncommutative Kähler structures were recently introduced as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant Kähler structures of irreducible type (those having an irreducible space of holomorphic 1-forms) we provide simple cohomological criteria for positivity, allowing one to avoid explicit curvature calculations. These general results are applied to our motivating family of examples, the irreducible quantum flag manifolds $\mathcal {O}_q(G/L_S)$. Building on the recently established noncommutative Borel–Weil theorem, every relative line module over $\mathcal {O}_q(G/L_S)$ can be identified as positive, negative, or flat, and it is then concluded that each Kähler structure is of Fano type.

Appendices

Appendix A: Some categorical equivalences

In this appendix, we present a number of categorical equivalences, all ultimately derived from Takeuchi’s equivalence [48]. These equivalences play a prominent role in the paper, giving us a formal framework in which to understand covariant differential calculi as relative Hopf modules.

1.1 A.1 Takeuchi’s equivalence

This subsection concisely presents Takeuchi’s equivalence for relative Hopf modules, as originally established in [48], and developed in [39]. We also present the monoidal version, as considered in [41, §4], and then restrict to the finitely generated subcategory of relative Hopf modules, as considered, for example, in [42, Corollary 2.5].

Let $\pi : A \rightarrow H$ be a surjective Hopf algebra map between Hopf algebras A and H. Then, a homogeneous right H-coaction is given by the map $\Delta _R:= (\textrm{id}\otimes \pi ) \circ \Delta : A \rightarrow A \otimes H.$ The associated quantum homogeneous space is defined to be the space of coinvariant elements, $A^{\textrm{co}(H)}$, that is,

$$\begin{aligned} A^{\textrm{co}(H)}:= \{ a \in A \mid \Delta _R (a) = a \otimes 1 \}.\end{aligned}$$

For any quantum homogeneous space $B = A^{\textrm{co}(H)}$, we define $^A_B\textrm{Mod}_B$ to be the category whose objects are left A-comodules $\Delta _L:\mathcal {F} \rightarrow A \otimes \mathcal {F}$, endowed with a B-bimodule structure such that $\Delta _L(bfc) = \Delta _L(b)\Delta _L(f)\Delta (c)$, for all $f \in \mathcal {F}, \, b,c \in B$, and whose morphisms are left A-comodule, B-bimodule, maps.

Let $^H\!\textrm{Mod}_B$ denote the category of relative Hopf modules, that is, the category whose objects are left H-comodules $\Delta _L: V \rightarrow H \otimes V$, endowed with a right B-module structure, such that $\Delta _L(vb) = v_{(-1)}\pi (b_{(1)}) \otimes v_{(0)}b_{(2)}$, for all $v \in V, \, b \in B$, and whose morphisms are left H-comodule maps and right B-module maps.

Consider the functor $\Phi :{}^A_B\textrm{Mod}_B \rightarrow {}^{H}\textrm{Mod}_B$, given by $\Phi (\mathcal {F}):= \mathcal {F}/B^+\mathcal {F}$, where the left H-comodule structure of $\Phi (\mathcal {{{\mathcal {F}}}})$ is given by $\Delta _L[f]:= \pi (f_{(-1)})\otimes [f_{(0)}],$ with square brackets denoting the coset of an element in $\Phi (\mathcal {{{\mathcal {F}}}})$. In the other direction, we define a functor $\Psi : {}^{H}\textrm{Mod}_B \rightarrow {}^A_B\textrm{Mod}_B$ by setting $\Psi (V):= A \,\square _{H} V$, where the left A-comodule structures of $\Psi (V)$ is defined on the first tensor factor, the right B-module structure is the diagonal one, and if $\gamma$ is a morphism in ${}^{H}\textrm{mod}_B$, then $\Psi (\gamma ):= \textrm{id}\otimes \gamma$.

An adjoint equivalence of categories between ${}^A_B\textrm{Mod}_B$ and ${}^{H}\textrm{Mod}_B$, which we call Takeuchi’s equivalence, is given by the functors $\Phi$ and $\Psi$, and the unit natural isomorphism

$$\begin{aligned} \textrm{U}: {{\mathcal {F}}}\rightarrow \Psi \circ \Phi ({{\mathcal {F}}}), \quad \quad f \mapsto f_{(-1)} \otimes [f_{(0)}]. \end{aligned}$$

The dimension $\textrm{dim}({{\mathcal {F}}})$ of an object ${{\mathcal {F}}}\in {}^A_B\textrm{Mod}$ is the vector space dimension of $\Phi ({{\mathcal {F}}})$. As observed in [41, Corollary 2.7], the inverse of the unit $\textrm{U}$ of the equivalence admits a useful explicit description:

$$\begin{aligned} \textrm{U}^{-1}\!\left( \sum _i f_i \otimes [g_i] \right) = \sum _i f_iS\big ((g_i)_{(-1)}\big )(g_i)_{(0)}. \end{aligned}$$

(12)

Consider $^A_B\text {Mod}_0$ the full subcategory of ${}^A_B\textrm{Mod}_B$ whose objects ${{\mathcal {F}}}$ satisfy $B^+{{\mathcal {F}}}= {{\mathcal {F}}}B^+$. The corresponding full subcategory ${}^H\textrm{Mod}_0$ of ${}^H\textrm{Mod}_B$ is given by objects with the trivial right B-action. The category $^A_B\text {Mod}_0$ comes equipped with a monoidal structure given by the tensor product $\otimes _B$. Moreover, with respect to the obvious monoidal structure on ${}^H\textrm{Mod}_0$, Takeuchi’s equivalence is readily endowed with the structure of a monoidal equivalence (see [41, §4]). An immediate implication is that an object ${{\mathcal {E}}}$ is invertible (that is, it is a relative line module) if and only if $\dim ({{\mathcal {E}}}) = 1$.

Finally, we consider $^A_B\text {mod}_0$ the full subcategory of ${}^A_B\textrm{Mod}_0$ whose objects are finitely generated as left B-modules, and note that it is a monoidal subcategory of ${}^A_B\textrm{Mod}_0$. The corresponding full subcategory ${}^H\textrm{mod}$ of ${}^H\textrm{Mod}$ has as objects the finite-dimensional left H-comodules, and its monoidal structure is the usual tensor product of comodules.

1.2 A.2 Conjugates and duals

Let us now assume that A and H are Hopf $*$-algebras, and that $\pi :A \rightarrow H$ is a Hopf $*$-algebra map. For any relative Hopf module ${{\mathcal {F}}}$, let $\overline{{{\mathcal {F}}}}$ be the relative Hopf module whose bimodule structure is defined by $b{\overline{f}}c = \overline{c^*fb^*}$, and whose left A-comodule structure is defined by $\Delta _L({\overline{f}}):= (f_{(-1)})^* \otimes \overline{f_{(0)}}$. Note that as a right B-module, $\overline{{{\mathcal {F}}}}$ is isomorphic to the conjugate of ${{\mathcal {F}}}$ as defined in 2.5, justifying the choice of notation. As shown in [43, Corollary 2.11], if ${{\mathcal {F}}}\in \, ^A_B\text {Mod}_0$, then $\overline{{{\mathcal {F}}}} \in \, ^A_B\text {Mod}_0$. It is instructive to note that the corresponding operation, through Takeuchi’s equivalence, on any object in $V \in \,^H\text {Mod}_0$ is the usual complex conjugate of V coming from the Hopf $*$-algebra structure of H.

We now restrict to the categories $^A_B\textrm{mod}_0$ and $^H\textrm{mod}$, and discuss dual objects. Since $^H\textrm{mod}$ is a rigid monoidal category, $^A_B\textrm{mod}$, or equivalently $^A_B\textrm{mod}_0$, is a rigid monoidal category. In particular, every object ${{\mathcal {F}}}\in \, ^A_B\textrm{mod}_0$ admits a dual object, which we denote by ${}^{\vee }\!{{\mathcal {F}}}$. Moreover, as shown in [43, Appendix A], by extending its usual bimodule structure, we can give $\, _{B}\textrm{Hom}({{\mathcal {F}}},B)$ the structure of an object in $\,^A_B\textrm{Mod}_B$, with respect to which it is right dual to ${{\mathcal {F}}}$. Now $^A_B\textrm{mod}_0$ is a monoidal subcategory of $\,^A_B\textrm{Mod}_B$. Thus, since right duals are unique up to unique isomorphism, ${}^{\vee }\!{{\mathcal {F}}}$ must be isomorphic to $\, _{B}\textrm{Hom}({{\mathcal {F}}},B)$, justifying the abuse of notation. Finally, we note that if H is a CQGA, then for any $V \in \,^H\textrm{mod}_0$, its dual and conjugate are always isomorphic, see [34, Theorem 11.27] for details.

Appendix B: A remark on the definition of the quantum Levi subgroup

In this subsection, we show the image of the restriction map $\pi _S$ defined in §4.3 is the type-1 dual of $U_q({\mathfrak {l}}_S)$, for any subset of simple nodes of a semisimple Lie algebra ${\mathfrak {g}}$. In fact, this is taken as the definition of $\mathcal {O}_q(L_S)$ in [18], and is the definition used in [16, 25, 26]. This result is a basic exercise in representation theory, and can be directly concluded from [27, Lemma 2.1]. We include a proof for the reader’s convenience.

Proposition B.1

Let S be a subset of the simple roots $\Pi$ of ${\mathfrak {g}}$, then the Hopf $*$-algebra map $\pi _S:\mathcal {O}_q(G) \rightarrow \mathcal {O}_q(L_S)$ is surjective.

Proof

Take an arbitrary finite-dimensional type-1 irreducible $U_q({\mathfrak {g}})$-module V and decompose it into irreducible type-1 $U_q({\mathfrak {l}}_S)$-submodules:

$$\begin{aligned} V \simeq \bigoplus _{\mu \, \in \mathcal {P}^+ \cup \mathcal {P}_{S^c}} W_{\mu }. \end{aligned}$$

This gives a corresponding coordinate coalgebra decomposition

$$\begin{aligned} \pi _S(C(V_{\lambda })) \simeq \bigoplus _{\mu \in \mathcal {P}^+ \cup \mathcal {P}_{S^c}} C(W_{\mu }). \end{aligned}$$

(13)

Thus, we see that the image of $\pi _S$ is contained in $\mathcal {O}_q(L_S)$, the type-1 dual of $U_q({\mathfrak {l}}_S)$.

To prove surjectivity, we need to show that for every weight $\nu \in \mathcal {P}^+ \cup \mathcal {P}_{S^c}$, the coordinate algebra $C(W_{\nu })$ is contained in the image of $\pi _S$. By (13), this would follow from a demonstration that every $W_{\nu }$ appears as a $U_q({\mathfrak {l}}_S)$-submodule of some $U_q({\mathfrak {g}})$-module.

Every element of $\mathcal {P}^+ \cup \mathcal {P}_{S^c}$ is a sum of weights of the form $\lambda + \mu$, where

$$\begin{aligned} \lambda \in \mathcal {P}^+, \quad \quad \text { and } \quad \quad -\mu \in \mathcal {P}_{S^c} \cap \mathcal {P}^+. \end{aligned}$$

Choose a highest weight vector $v_{\textrm{hw}}$ in the irreducible $U_q({\mathfrak {g}})$-module $V_{\lambda }$, and choose a lowest weight vector $v_{\textrm{lw}}$ in the irreducible $U_q({\mathfrak {g}})$-module $V_{w_0(\mu )}$. We see that since

$$\begin{aligned} v_{\textrm{hw}} \otimes v_{\textrm{lw}} \in V_{\lambda } \otimes V_{\mu } \end{aligned}$$

is a $U_q({\mathfrak {l}}_S)$-highest weight vector, $U_q({\mathfrak {l}}_S)v_{\textrm{hw}} \otimes v_{\textrm{lw}}$ is an irreducible $U_q({\mathfrak {l}}_S)$-submodule of $V_{\lambda } \otimes V_{w_0(\mu )}$ of highest weight $\lambda + \mu$. Thus, we see that $C(W_{\lambda +\mu })$ is contained in the image of $\pi _S$, and hence that $\pi _S$ is surjective. $\square$

Appendix C: Tables for the irreducible quantum flag manifolds

We recall the standard pictorial description of the quantum Levi subalgebras defining the irreducible quantum flag manifolds, given in terms of Dynkin diagrams.

For a diagram of rank r, to the black node $\alpha _x$ we associate the set $S:= \{\alpha _1,\ldots ,\alpha _r\}\backslash \{\alpha _x\}$, with corresponding Levi subgroup $L_S$. The irreducible quantum flag manifold is then given by the coinvariant subspace $\mathcal {O}_q(G/L_S) \subseteq \mathcal {O}_q(G)$. Note that any automorphism of a Dynkin diagram results in an isomorphic quantum flag manifold, which is not denoted in the diagram. In particular, for the case of $D_n$ and $E_6$, colouring the second spinor node, and the first node, respectively, produces an isomorphic copy of the corresponding quantum flag manifold.

We present an explicit description of the canonical line modules (see Table 2) of the irreducible quantum flag manifolds using the approach of Theorem 4.12. All line modules are indexed by the negative integers, and hence are negative in the sense of Definition 2.7. The values coincide with their classical counterparts, see, for example, [31, § II.4]. This allows us to conclude in Theorem 4.12 that the Kähler structure of each irreducible quantum flag manifold is of Fano type.

Table 1 Irreducible Quantum Flag Manifolds: organised by series, with defining crossed node numbered according to [28, §11.4], CQGA-homogeneous space symbol and name

Table 2 Irreducible Quantum Flag Manifolds: CQGA-homogeneous space symbol, corresponding Heckenberger–Kolb calculus complex dimension, and the space of top holomorphic forms identified as a line module

Remark C.1

By a theorem of Atiyah, a 2m-dimensional compact Hermitian manifold is spin if and only if its canonical line module $\Omega ^{(m,0)}$ admits a holomorphic square root [2, Proposition 3.2]. Thus from Table 2, we see that the classical Grassmannians $\textrm{Gr}_{s,n+1}$, and the classical Lagrangian Grassmannians $\textbf{L}_n$, are spin for all $n \in 2{\mathbb {Z}}_{> 0} + 1$. Moreover, the even quadrics $\textbf{Q}_{2n}$, and the spinor varieties $\textbf{S}_n$, are spin, for all $n \in {\mathbb {Z}}_{> 0}$. For the exceptional cases, both the Cayley plane and the Freudenthal variety are spin. Atiyah’s theorem suggests a definition for noncommutative Hermitian spin structures with a substantial ready-made family of noncommutative examples.