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Register a new userQuantized enveloping superalgebra of type $P$
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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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Given a symmetric operad $mathcal{P}$ and a $mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${mathsf{U}_{mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad. In case $mathcal{P}$ is Koszul a criterion for the PBW property is found. A necessary condition on the Hilbert series for $mathcal{P}$ is discovered. Moreover, given any symmetric operad $mathcal{P}$, together with a Grobner basis $G$, a condition is given in terms of the structure of the underlying trees associated with leading monomials of $G$, sufficient for the PBW property to hold. Examples are provided.
We give explicit actions of Drinfeld generators on Gelfand-Tsetlin bases of super Yangian modules associated with skew Young diagrams. In particular, we give another proof that these representations are irreducible. We study irreducible tame $mathrm Y(mathfrak{gl}_{1|1})$-modules and show that a finite-dimensional irreducible $mathrm Y(mathfrak{gl}_{1|1})$-module is tame if and only if it is thin. We also give the analogous statements for quantum affine superalgebra of type A.
Let $frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(frak{g}))$ of the quantum group $U_q(frak{g})$ is isomorphic to a monoid algebra, and that $Z(U_q(frak{g}))$ is a polynomial algebra if and only if $frak{g}$ is of type $A_1, B_n, C_n, D_{2k+2}, E_7, E_8, F_4$ or $G_2.$ Moreover, in case $frak{g}$ is of type $D_{n}$ with $n$ odd, then $Z(U_q(frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in $n+1$ variables with one relation; in case $frak{g}$ is of type $E_6$, then $Z(U_q(frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in fourteen variables with eight relations; in case $frak{g}$ is of type $A_{n}$, then $Z(U_q(frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra described by $n$-sequences.
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 $kappa$.
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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