Howe duality for quantum queer superalgebras

Abstract

We establish a new Howe duality between a pair of quantum queer superalgebras (U q −1 (qn), Uq(qm)). The key ingredient is the construction of a non-commutative analogue Aq(qn, qm) of the symmetric superalgebra S(C mn|mn ) with the use of quantum coordinate queer superalgebra. It turns out that this superalgebra is equipped with a U q −1 (qn) ⊗ Uq(qm)supermodule structure that admits a multiplicity-free decomposition. We also show that the (U q −1 (qn), Uq(qm))-Howe duality implies the Sergeev-Olshanski duality.MSC(2010): 17B37, 17D10, 20G42.

Webs of type Q

“…A straightforward computation using (12) verifies the following. S λ be the decomposition into weight spaces with respect to the Cartan subalgebra of q(m).…”

“…In other work the first author uses the category q-Web ↑ to give a complete combinatorial description of the full subcategory of permutation supermodules for the Sergeev algebra [3]. In [12] Chang-Wang introduce a quantum Howe duality of type Q. This is done by defining an action of quantized enveloping algebras of type Q on a quantized symmetric algebra which also appeared in [2].…”

Some properties for almost cellular algebras

“…Motivated by Schur-Weyl duality, the authors introduce the quantum walled Brauer-Clifford superalgebras BC r,s (q), the quantum deformation of the walled Brauer-Clifford superalgebra BC r,s , which is the centralizer superalgebra of the action of U q (q(n)) on the mixed tensor space under some mild condition [1]. The Howe duality for quantum queer superalgebras was given [2]. Because of their similarity with Hecke-Clifford (super)algebras, the quantum walled Brauer-Clifford (super)algebras are not the cellular algebras since the absence of an anti-involution with the property that it fixes isomorphism classes of irreducible modules.…”

Geometric Howe dualities of finite type

“…The Howe duality for a pair of quantum general linear groups U q (gl m ) and U q (gl n ) was given by Zhang in [Z02], where quantum coordinate algebras were employed to construct a non-commutative analogue of the symmetric algebras on which U q (gl m ) and U q (gl n ) act. This construction was further applied to established the Howe duality of (U q (gl n ), U q (so 2n )), (U q (gl n ), U q (so 2n+1 )) and (U q (gl n ), U q (sp 2n )) in [LZ03] (see also [WZ09,CW20] for quantum supergroups). It also helps to provide a non-commutative version of the FFT for associated quantum groups (cf.…”