This paper provides a realization of \(\imath\)quantum groups as Hall algebras.
In the first part, the authors construct semi-derived Ringel-Hall algebras of quivers with involutions, called \(\imath\)quivers. To an \(\imath\)quiver \((Q,\tau)\) they associate a finite-dimensional algebra \(\Lambda^\imath\), and explicitly describe \(\Lambda^\imath\) as a bound quiver algebra as well as a tensor algebra. It is shown that \(\Lambda^\imath\) is \(1\)-Gorenstein, which is necessary for the construction of the (twisted) semi-derived Ringel-Hall algebras \(\mathcal{SDH}(\Lambda^\imath)\) and \(\mathcal{SD}\widetilde{\mathcal{H}}(\Lambda^\imath)\), and the reduced version \(\mathcal{SDH}_{\mathrm{red}}(\Lambda^\imath)\). This construction and the theory of semi-derived Ringel-Hall algebras in general are explained in an appendix by the first author. From the general theory, the authors in particular obtain PBW bases and monomial bases of \(\mathcal{SDH}(\Lambda^\imath)\) and \(\mathcal{SD}\widetilde{\mathcal{H}}(\Lambda^\imath)\) for \(Q\) of Dynkin type.
The involution \(\tau\) also induces an involution \(\hat{\tau}\) on \(D^b(kQ)\), and the authors prove that the singularity category \(D_{\mathrm{sg}}(\Lambda^\imath)\) is equivalent to \(D^b(kQ)/\Sigma\hat{\tau}\). By a well-known theorem of Buchweitz and Happel, these categories can moreover be identified with the stable category of Gorenstein projective \(\Lambda^\imath\)-modules. Furthermore, it is shown that \(\mathcal{SDH}(\Lambda^\imath)\) is isomorphic to \(\mathcal{SDH}(\mathrm{End}(T))\) for any tilting module \(T\) such that \(\mathrm{End}(T)\) is \(1\)-Gorenstein.
From an acyclic \(\imath\)quiver one obtains the quantum symmetric pair \((\mathbf{U},\mathbf{U}^\imath)\) consisting of the quantum group and the \(\imath\)quantum group, a certain coideal subalgebra. The authors introduce a universal version \((\widetilde{\mathbf{U}},\widetilde{\mathbf{U}}^\imath)\) using the universal quantum group \(\widetilde{\mathbf{U}}\), which in comparison to \(\mathbf{U}\) has the Cartan subalgebra doubled.
In the second part of the paper, the authors construct isomorphisms of algebras \(\widetilde{\mathbf{U}}^\imath_{|v=\sqrt{q}}\cong \mathcal{SD}\widetilde{\mathcal{H}}(\Lambda^\imath)\) and \(\mathbf{U}^\imath_{|v=\sqrt{q}}\cong\mathcal{SDH}_{\mathrm{red}}(\Lambda^\imath)\) for Dynkin \(\imath\)quivers, with \(\Lambda^\imath\) considered over the finite field \(\mathbb{F}_q\). Furthermore, they obtain embeddings of algebras \(\widetilde{\mathbf{U}}_{|v=\sqrt{q}}\hookrightarrow\mathcal{SD}\widetilde{\mathcal{H}}(\Lambda)\) and \(\mathbf{U}_{|v=\sqrt{q}}\hookrightarrow\mathcal{SDH}_{\mathrm{red}}(\Lambda)\), where \(\Lambda=(\Lambda^{\mathrm{dbl}})^\imath\) is constructed by doubling the quiver \(Q\). These embeddings are isomorphisms if and only if \(Q\) is of Dynkin type. This is a version of a result of Bridgeland [T. Bridgeland, Ann. Math. (2) 177, No. 2, 739–759 (2013; Zbl 1268.16017)].
Moreover, the authors show that the structure constants of \(\mathcal{SD}\widetilde{\mathcal{H}}(\Lambda^\imath)\) and \(\mathcal{SDH}_{\mathrm{red}}(\Lambda^\imath)\) are Laurent polynomials, and use this to construct the (reduced) generic Hall algebras \(\widetilde{\mathcal{H}}(Q,\tau)\) and \(\mathcal{H}_{\mathrm{red}}(Q,\tau)\). By construction, these specialize to the (twisted respectively reduced) semi-derived Ringel-Hall algebras, and it follows that for Dynkin quivers the (reduced) generic Hall algebras are isomorphic to \(\widetilde{\mathbf{U}}^\imath\) and \(\mathbf{U}^\imath\), respectively. Altogether, these results in particular provide PBW bases for \(\widetilde{\mathbf{U}}^\imath\) and \(\mathbf{U}^\imath\).
The paper is the first in a series of three articles, for Parts II and III see [M. Lu and W. Wang, Commun. Math. Phys. 381, No. 3, 799–855 (2021; Zbl 1479.17029)] and [M. Lu and W. Wang, Adv. Math. 393, Article ID 108071, 70 p. (2021; Zbl 1484.17025)].