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In this paper, we establish explicit relationship between decomposition numbers of quantized walled Brauer algebras and those for either Hecke algebras associated to certain symmetric groups or (rational) $q$-Schur algebras over a field $kappa$. This enables us to use Arikis result cite{Ar} and Varagnolo-Vasserots result cite{VV} to compute such decomposition numbers via inverse Kazhdan-Lusztig polynomials associated with affine Weyl groups of type $A$ if the ground field is $mathbb C$.
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In this paper, we give a criterion on the semisimplicity of quantized walled Brauer algebras $mathscr B_{r,s}$ and classify its simple modules over an arbitrary field $kappa$.
Motivated by Brundan-Kleshchevs work on higher Schur-Weyl duality, we establish mixed Schur-Weyl duality between general linear Lie algebras and cyclotomic walled Brauer algebras in an arbitrary level. Using weakly cellular bases of cyclotomic walled Brauer algebras, we classify highest weight vectors of certain mixed tensor modules of general linear Lie algebras. This leads to an efficient way to compute decomposition matrices of cyclotomic walled Brauer algebras arising from mixed Schur-Weyl duality, which generalizes early results on level two walled Brauer algebras.
In this paper, a notion of affine walled Brauer-Clifford superalgebras $BC_{r, t}^{rm aff} $ is introduced over an arbitrary integral domain $R$ containing $2^{-1}$. These superalgebras can be considered as affinization of walled Brauer superalgebras in cite{JK}. By constructing infinite many homomorphisms from $BC_{r, t}^{rm aff}$ to a class of level two walled Brauer-Clifford superagebras over $mathbb C$, we prove that $BC_{r, t}^{rm aff} $ is free over $R$ with infinite rank. We explain that any finite dimensional irreducible $BC_{r, t}^{rm aff} $-module over an algebraically closed field $F$ of characteristic not $2$ factors through a cyclotomic quotient of $BC_{r, t}^{rm aff} $, called a cyclotomic (or level $k$) walled Brauer-Clifford superalgebra $ BC_{k, r, t}$. Using a previous method on cyclotomic walled Brauer algebras in cite{RSu1}, we prove that $BC_{k, r, t}$ is free over $R$ with super rank $(k^{r+t}2^{r+t-1} (r+t)!, k^{r+t}2^{r+t-1} (r+t)!)$ if and only if it is admissible in the sense of Definition~6.4. Finally, we prove that the degenerate affine (resp., cyclotomic) walled Brauer-Clifford superalgebras defined by Comes-Kujawa in cite{CK} are isomorphic to our affine (resp., cyclotomic) walled Brauer-Clifford superalgebras.
Let L be a Lie pseudoalgebra, a in L. We show that, if a generates a (finite) solvable subalgebra S=<a>, then one may find a lifting a in S of [a] in S/S such that <a> is nilpotent. We then apply this result towards vertex algebras: we show that every finite vertex algebra V admits a decomposition into a semi-direct product V = U + N, where U is a subalgebra of V whose underlying Lie conformal algebra U^lie is a nilpotent self-normalizing subalgebra of V^lie, and N is a canonically determined ideal contained in the nilradical Nil V.
We introduce a new quantized enveloping superalgebra $mathfrak{U}_q{mathfrak{p}}_n$ attached to the Lie superalgebra ${mathfrak{p}}_n$ of type $P$. The superalgebra $mathfrak{U}_q{mathfrak{p}}_n$ is a quantization of a Lie bisuperalgebra structure on ${mathfrak{p}}_n$ and we study some of its basic properties. We also introduce the periplectic $q$-Brauer algebra and prove that it is the centralizer of the $mathfrak{U}_q {mathfrak{p}}_n$-module structure on ${mathbb C}(n|n)^{otimes l}$. We end by proposing a definition for a new periplectic $q$-Schur superalgebra.
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