Abstract
Quaternionic projective plane \(\mathbb {H} P^{2}\) is the next simplest conjugacy class of a complex symplectic group with pseudo-Levi stabilizer subgroup after the sphere \(\mathbb {S}^{4}\simeq \mathbb {H} P^{1}\). Its quantization gives rise to a module category \(\mathcal {O}_{t}\bigl (\mathbb {H} P^{2}\bigr )\) over finite-dimensional representations of the symplectic quantum group \(U_{q}\bigl (\mathfrak {s}\mathfrak {p}(6)\bigr )\), a full subcategory in the BGG category \(\mathcal {O}\). We prove that \(\mathcal {O}_{t}\bigl (\mathbb {H} P^{2}\bigr )\) is semi-simple and equivalent to a category of quantized equivariant vector bundles on \(\mathbb {H} P^{2}\).
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Acknowledgements
The second author (AM) is grateful to Steklov Mathematical Institute, St.-Petersburg Department for a support from a grant for creation and development of International Mathematical Centers, agreement no. 075-15-2019-16-20 of November 8, 2019, between Ministry of Science and Higher Education of Russia and PDMI RAS. This work was also partially supported by the Moscow Institute of Physics and Technology under the Priority 2030 Strategic Academic Leadership Program.
The authors are indebted to the anonymous referee whose careful reading of the manuscript and valuable comments and suggestions greatly helped us to improve and extend the original text.
Funding
Author A. Mudrov received partial research support from a grant for creation and development of International Mathematical Centers, agreement no. 075-15-2019-16-20 of November 8, 2019, between Ministry of Science and Higher Education of Russia and PDMI RAS.
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Appendix
Appendix
In this technical section, we derive some identities in the algebra \(U_{q}(\mathfrak {g}_{-})\) which are needed for this exposition.
Lemma A.1
Define \(\bar {f}_{\theta }\) obtained from f𝜃 by replacement \(q\to \bar {q}\). Then
Proof
We will use a modified Jacobi identity
which holds true for any elements x,y,z of an associative algebra and any scalars a,b,c with invertible c. This can be verified by a direct calculation.
Now let us prove the right equality in (A.15). Apply (A.17) to \([f_{2},[[f_{1},f_{2}]_{\bar {q}},f_{3}]_{\bar {q}^{2}}]_{q}\) choosing \(c=\bar {q}\):
The first summand vanishes thanks to the Serre relation of weight − (2α2 + α1) whence (A.15) follows. Then (A.16) follows from (A.15) by replacement \(q\to \bar q\). □
Lemma A.2
One has
Proof
Apply (A.17) to \([f_{1},f_{\delta }]=[f_{1},[f_{2},[f_{2},f_{3}]_{q^{2}}]_{\bar q^{2}}]\) choosing \(c=\bar q\). Then
The first summand is qf𝜃 from (A.15). In the second summand, replace \([f_{1},[f_{2},f_{3}]_{q^{2}}]_{q}\) with \([[f_{1},f_{2}]_{q},f_{3}]_{q^{2}}\), then it becomes \(\bar q \bar f_{\theta }\) from (A.18). □
Other identities of interest can be also derived from the Serre relations a with the use of the modified Jacobi identity (A.17). We will give another proof based on Lusztig’s braid group automorphisms of \(U_{q}(\mathfrak {g})\), [2].
Proposition A.3
The following relations hold true in \(U_{q}(\mathfrak {g}_{-})\):
where \(f_{\nu }=[f_{1},f_{2}]_{\bar q}\).
Proof
Let Ti be Lusztig automorphisms of \(U_{q}(\mathfrak {g})\) corresponding to simple reflections σi: R →R relative the simple roots αi, as in [2]. They satisfy braid group relations, of which we will need only
In particular, \(f_{\nu }=T^{-1}_{2}(f_{1})\) and \(T^{-1}_{3}(f_{2})=[f_{2},f_{3}]_{\bar q^{2}}\) which implies
because \(T^{-1}_{3}(f_{1})=f_{1}\). Set \(w=T^{-1}_{3}T^{-1}_{2}T^{-1}_{3}\), then
The first equality has been checked. The second equality is fulfilled because σ3σ2σ3(α2) = α2. The third formula follows from the first two as w is an algebra automorphism. The last one readily follows from the equality \(T_{2}^{-1} T_{3}^{-1} T_{2}^{-1}(e_{3})=e_{3}\) as a result of \(T^{-1}_{3}(e_{3})\), cf. [2].
Applying \(wT^{-1}_{2}\) to a commuting pair (e3,f1) one gets the left equality (A.19) because (𝜃,α3) = 0. Applying w to a quasi-commuting pair (f2,fν), one gets the left equality in (A.20). The right equalities in (A.19) and (A.20) result from replacement q → q− 1. Then (A.20) follows since fδ comprises two f2-factors and one f3-factor. To prove (A.22), apply \(T^{-1}_{2}T^{-1}_{3}\) to a quasi-commuting pair of f1 and \(f_{\nu }\simeq T^{-1}_{2} T^{-1}_{3}(f_{1})\), using the equality \(T^{-1}_{3} (f_{1})=f_{1}\) and the braid relation. The formula (A.23) is obtained by applying w to quasi-commuting f2 and fν. □
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Jones, G., Mudrov, A. Pseudo-parabolic Category over Quaternionic Projective Plane. Algebr Represent Theor 26, 2361–2382 (2023). https://doi.org/10.1007/s10468-022-10185-8
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DOI: https://doi.org/10.1007/s10468-022-10185-8