advanced_tools:group_theory:representation_theory:adjoint_representation
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advanced_tools:group_theory:representation_theory:adjoint_representation [2020/12/26 22:49] edi [Concrete] |
advanced_tools:group_theory:representation_theory:adjoint_representation [2023/03/19 21:41] edi [Concrete] |
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| - | The adjoint representation describes how the generators of the group, which live in the Lie algebra, transform. In other words, the adjoint representation acts on the Lie algebra. Note that the Lie algebra is a regular vector space and thus can serve as a representation space. | + | The adjoint representation describes how the generators of the group, which live in the Lie algebra, transform. In other words, the adjoint representation acts on the Lie algebra. Note that the Lie algebra is a vector space and thus can serve as a representation space. |
| The dimensionality of the adjoint representation is equal to that of the Lie algebra. For example, the adjoint representation of $SU(2)$ is 3 dimensional and the adjoint representation of $SU(3)$ is 8 dimensional. | The dimensionality of the adjoint representation is equal to that of the Lie algebra. For example, the adjoint representation of $SU(2)$ is 3 dimensional and the adjoint representation of $SU(3)$ is 8 dimensional. | ||
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| [{{ :advanced_tools:group_theory:representation_theory:su2_adjoint.jpg?nolink }}] | [{{ :advanced_tools:group_theory:representation_theory:su2_adjoint.jpg?nolink }}] | ||
| - | For a more detailed explanation of this diagram as well as adjoint representations of other groups, see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. | + | For a more detailed explanation of this diagram as well as adjoint representations of other groups, see [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#su2_adjoint|Fun with Symmetry]]. |
| <tabbox Abstract> | <tabbox Abstract> | ||
advanced_tools/group_theory/representation_theory/adjoint_representation.txt ยท Last modified: 2023/03/19 21:41 by edi
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