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Added coroot description of $Z$
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Victor Protsak
Victor Protsak
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By the highest weight theory, finite-dimensional irreducible representations of a finite-dimensional simple Lie algebra $\mathfrak{g}$ are parametrized by the dominant weights $\lambda$ in the weight lattice $Q$. On the other hand, the analogous representations of the adjoint form $G_{\rm ad}$ of the corresponding simple Lie group are parametrized by the dominant weights in the root lattice $P$. The difference is measured by the finite abelian group $Q/P$, which may be identifed with the dual of the center $Z$ of the simply-connected form $G_{\rm sc}$ (in fact, $Z\simeq\! P^\vee/Q^\vee$). A representation of $G_{\rm sc}$ is faithful if and only if its restriction to $Z$ is faithful if and only if $\lambda$ is not contained in any proper sublattice of $Q$.

This explained and tabulated in terms of the corresponding root systems in nearly all introductory Lie theory textbooks, but I am partial to exposition in Goto and Grosshans, where I first learned it.

By the highest weight theory, finite-dimensional irreducible representations of a finite-dimensional simple Lie algebra $\mathfrak{g}$ are parametrized by the dominant weights $\lambda$ in the weight lattice $Q$. On the other hand, the analogous representations of the adjoint form $G_{\rm ad}$ of the corresponding simple Lie group are parametrized by the dominant weights in the root lattice $P$. The difference is measured by the finite abelian group $Q/P$, which may be identifed with the dual of the center $Z$ of the simply-connected form $G_{\rm sc}$. A representation of $G_{\rm sc}$ is faithful if and only if its restriction to $Z$ is faithful if and only if $\lambda$ is not contained in any proper sublattice of $Q$.

This explained and tabulated in terms of the corresponding root systems in nearly all introductory Lie theory textbooks, but I am partial to exposition in Goto and Grosshans, where I first learned it.

By the highest weight theory, finite-dimensional irreducible representations of a finite-dimensional simple Lie algebra $\mathfrak{g}$ are parametrized by the dominant weights $\lambda$ in the weight lattice $Q$. On the other hand, the analogous representations of the adjoint form $G_{\rm ad}$ of the corresponding simple Lie group are parametrized by the dominant weights in the root lattice $P$. The difference is measured by the finite abelian group $Q/P$, which may be identifed with the dual of the center $Z$ of the simply-connected form $G_{\rm sc}$ (in fact, $Z\simeq\! P^\vee/Q^\vee$). A representation of $G_{\rm sc}$ is faithful if and only if its restriction to $Z$ is faithful if and only if $\lambda$ is not contained in any proper sublattice of $Q$.

This explained and tabulated in terms of the corresponding root systems in nearly all introductory Lie theory textbooks, but I am partial to exposition in Goto and Grosshans, where I first learned it.

Source Link
Victor Protsak
Victor Protsak
  • 14.5k
  • 4
  • 68
  • 94

By the highest weight theory, finite-dimensional irreducible representations of a finite-dimensional simple Lie algebra $\mathfrak{g}$ are parametrized by the dominant weights $\lambda$ in the weight lattice $Q$. On the other hand, the analogous representations of the adjoint form $G_{\rm ad}$ of the corresponding simple Lie group are parametrized by the dominant weights in the root lattice $P$. The difference is measured by the finite abelian group $Q/P$, which may be identifed with the dual of the center $Z$ of the simply-connected form $G_{\rm sc}$. A representation of $G_{\rm sc}$ is faithful if and only if its restriction to $Z$ is faithful if and only if $\lambda$ is not contained in any proper sublattice of $Q$.

This explained and tabulated in terms of the corresponding root systems in nearly all introductory Lie theory textbooks, but I am partial to exposition in Goto and Grosshans, where I first learned it.