On Morita equivalences between KLR algebras and VV algebras (Q2188391)

\textit{M. Varagnolo} and \textit{E. Vasserot} [Invent. Math. 185, No. 3, 593--693 (2011; Zbl 1239.20007)] defined a new family of graded algebras, the representation theory of which is closely related to the representation theory of the affine Hecke algebras of type \(B\). They proved in [loc. cit.] that categories of finite-dimensional modules over these algebras are equivalent to categories of finite-dimensional modules over type \(B\) affine Hecke algebras, \(H_n^B\). They also utilize these algebras to prove a conjecture of \textit{N. Enomoto} and \textit{M. Kashiwara} which states that the representations of the affine Hecke algebra of type \(B\) categorify a simple highest weight module for a certain quantum group [Publ. Res. Inst. Math. Sci. 44, No. 3, 837--891 (2008; Zbl 1168.17008)]. We refer to these algebras as VV algebras. One of the advantages of working with VV algebras is that they have a non-trivial grading, while affine Hecke algebras of type \(B\) do not. Similarly to Khovanov-Lauda-Rouquier algebras (in brief, KLR algebras), VV algebras depend upon a quiver \(\Gamma\), which now comes equipped with an involution \(\theta\), and a dimension vector \(\nu\) which is invariant under this involution. The vertices of \(\Gamma\) are labelled by an indexing set \(I\) which is an orbit arising from a \(\mathbb{Z}\rtimes \mathbb{Z}_2\)-action on the ground field. They also depend upon two non-zero elements of the field, \(p\) and \(q\), which \(B\) correspond to the deformation parameters of \(H_n^B\) . It turns out that there are different cases to consider when studying VV algebras. In this article, the authors compare categories of modules over KLR algebras with categories of modules over VV algebras. In case of \(p, q\notin I\), they observe that these module categories are indeed equivalent. In other cases, they find categories of modules over VV algebras that are equivalent to categories of modules over the tensor product of KLR algebras with the path algebra of a certain quiver. Using these Morita equivalences, many properties of KLR algebras, and the given path algebra, can be transferred to classes of VV algebras. In particular, the main result of this work is that certain subclasses of VV algebras are affine cellular and even affine quasi-hereditary.