Quiver Hecke superalgebras

Supercategorification of quantum Kac–Moody algebras

“…In [12], motivated by the works of Brundan-Kleshchev [3], Kang, Kashiwara and Tsuchioka introduced a new family of graded superalgebras, called the quiver Hecke superalgebras, which is a super version of Khovanov-Lauda-Rouquier algebras. They also defined the notion of quiver Hecke-Clifford superalgebras and showed that these superalgebras are weakly Morita superequivalent to the corresponding quiver Hecke superalgebras.…”

Supercategorification of quantum Kac–Moody algebras II

“…Quiver Hecke superalgebras. In this subsection, we recall the definition of quiver Hecke superalgebras and their basic properties ( [KKT11]). We take a graded commutative ring k = n∈Z ≥0 k n as a base ring.…”

“…Naturally, our next goal is to find a super-version of Khovanov-Lauda-Rouquier categorification theorem and Kang-Kashiwara cyclotomic categorification theorem. In [KKT11], Kang, Kashiwara and Tsuchioka introduced the notion of quiver Hecke superalgebras and quiver Hecke-Clifford superalgebras which are Z-graded algebras over a commutative graded ring k = ⊕ n≥0 k n with k 0 a field. They showed that these superalgebras are weakly Morita superequivalent and that, after some completion, the quiver Hecke-Clifford superalgebras are isomorphic to the affine Hecke-Clifford superalgebras.…”

“…Based on the results of [KKT11], Kang, Kashiwara and Oh proved that the quiver Hecke superalgebras and the cyclotomic quiver Hecke superalgebras provide a supercategorification of quantum Kac-Moody algebras and their integrable highest weight modules [KKO12]. Here, a supercategorification of an algebraic structure means a construction of a 1-supercategory or a 2-supercategory whose Grothendieck group is isomorphic to the given algebraic structure.…”

Quiver Hecke algebras and categorification

“…Even more variations on the quiver Hecke algebras have subsequently emerged, including a twisted version related to affine Hecke algebras of type B introduced by Varagnolo and Vasserot [33], and the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka [16]. The latter superalgebras generalize Wang's spin Hecke algebras [34] and the odd nil Hecke algebra of Ellis, Khovanov and Lauda [10], and give a completely new "supercategorification" of the same quantum groups/highest weight modules as above (see [15]).…”