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Register a new userA quantized Tits-Kantor-Koecher algebra
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We propose a quantum analogue of a Tits-Kantor-Koecher algebra with a Jordan torus as an coordinated algebra by looking at the vertex operator construction over a Fock space.
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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.
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.
We propose a general method to realize an arbitrary Weyl group of Kac-Moody type as a group of birational canonical transformations, by means of a nilpotent Poisson algebra. We also give a Lie theoretic interpretation of this realization in terms of Kac-Moody Lie algebras and Kac-Moody groups.
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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