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esteban
esteban
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Let $\mathbf{U}_q(\mathfrak{g})$ be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we use the Sweedler notation. The locally finite part of $\mathbf{U}_q(\mathfrak{g})$ is the subspace defined by $F_l(\mathbf{U}_q(\mathfrak{g}))=\{x\in \mathbf{U}_q(\mathfrak{g}): dim (ad(\mathbf{U}_q(\mathfrak{g}))(x)<\infty\}$$F_l(\mathbf{U}_q(\mathfrak{g}))=\{x\in \mathbf{U}_q(\mathfrak{g}): dim (ad(\mathbf{U}_q(\mathfrak{g}))(x))<\infty\}$. The locally finite part has a Peter-Weyl decomposition $$F_l(\mathbf{U}_q(\mathfrak{g}))=\bigoplus _{\lambda\in R} ad(\mathbf{U}_q(\mathfrak{g}))(K_{-\lambda}).$$ I wonder if the cyclic $ad$-modules $ad(\mathbf{U}_q(\mathfrak{g}))(K_{-\lambda})$ are invariant under the action of the Lusztig braid group operators.

Let $\mathbf{U}_q(\mathfrak{g})$ be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we use the Sweedler notation. The locally finite part of $\mathbf{U}_q(\mathfrak{g})$ is the subspace defined by $F_l(\mathbf{U}_q(\mathfrak{g}))=\{x\in \mathbf{U}_q(\mathfrak{g}): dim (ad(\mathbf{U}_q(\mathfrak{g}))(x)<\infty\}$. The locally finite part has a Peter-Weyl decomposition $$F_l(\mathbf{U}_q(\mathfrak{g}))=\bigoplus _{\lambda\in R} ad(\mathbf{U}_q(\mathfrak{g}))(K_{-\lambda}).$$ I wonder if the cyclic $ad$-modules $ad(\mathbf{U}_q(\mathfrak{g}))(K_{-\lambda})$ are invariant under the action of the Lusztig braid group operators.

Let $\mathbf{U}_q(\mathfrak{g})$ be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we use the Sweedler notation. The locally finite part of $\mathbf{U}_q(\mathfrak{g})$ is the subspace defined by $F_l(\mathbf{U}_q(\mathfrak{g}))=\{x\in \mathbf{U}_q(\mathfrak{g}): dim (ad(\mathbf{U}_q(\mathfrak{g}))(x))<\infty\}$. The locally finite part has a Peter-Weyl decomposition $$F_l(\mathbf{U}_q(\mathfrak{g}))=\bigoplus _{\lambda\in R} ad(\mathbf{U}_q(\mathfrak{g}))(K_{-\lambda}).$$ I wonder if the cyclic $ad$-modules $ad(\mathbf{U}_q(\mathfrak{g}))(K_{-\lambda})$ are invariant under the action of the Lusztig braid group operators.

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esteban
esteban
  • 141
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Action of Lusztig braid group operators on locally finite part

Let $\mathbf{U}_q(\mathfrak{g})$ be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we use the Sweedler notation. The locally finite part of $\mathbf{U}_q(\mathfrak{g})$ is the subspace defined by $F_l(\mathbf{U}_q(\mathfrak{g}))=\{x\in \mathbf{U}_q(\mathfrak{g}): dim (ad(\mathbf{U}_q(\mathfrak{g}))(x)<\infty\}$. The locally finite part has a Peter-Weyl decomposition $$F_l(\mathbf{U}_q(\mathfrak{g}))=\bigoplus _{\lambda\in R} ad(\mathbf{U}_q(\mathfrak{g}))(K_{-\lambda}).$$ I wonder if the cyclic $ad$-modules $ad(\mathbf{U}_q(\mathfrak{g}))(K_{-\lambda})$ are invariant under the action of the Lusztig braid group operators.