advanced_tools:group_theory:lie_algebras
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advanced_tools:group_theory:lie_algebras [2018/04/08 12:07] jakobadmin |
advanced_tools:group_theory:lie_algebras [2023/03/29 17:40] (current) 132.76.61.54 fur -> for |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | <note tip> | + | The Lie algebras consist of infinitesimally small transformations and are used in physics to describe [[basic_tools:symmetry|symmetries]]. Since arbitrarily large continuous transformations can be built up from tiny ones, almost everything that is important about continuous transformations is encoded in the infinitesimal ones. |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | |
| - | </note> | + | Formulated differently, Lie algebras describe infinitesimal symmetries. |
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| <tabbox Concrete> | <tabbox Concrete> | ||
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| ** Definition:** | ** Definition:** | ||
| - | <WRAP tip> For a matrix Lie group $G$ the corresponding Lie algebra $\mathfrak{g}$ can be defined as the set of matrices $X$ that yield $e^{tX} \in G$ für $t \in \mathbb{R}$.</WRAP> | + | <WRAP tip> For a matrix Lie group $G$ the corresponding Lie algebra $\mathfrak{g}$ can be defined as the set of matrices $X$ that yield $e^{tX} \in G$ for $t \in \mathbb{R}$.</WRAP> |
| **The Natural Product of a Lie Algebra:** | **The Natural Product of a Lie Algebra:** | ||
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| <WRAP tip> The natural product of a Lie algebra is not ordinary matrix multiplication, but the Lie bracket.</WRAP> | <WRAP tip> The natural product of a Lie algebra is not ordinary matrix multiplication, but the Lie bracket.</WRAP> | ||
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| - | * [[http://jakobschwichtenberg.com/lie-algebra-able-describe-group/|How is a Lie Algebra able to describe a Group?]] | ||
| - | * [[http://www.wetsavannaanimals.net/wordpress/what-information-about-a-lie-group-is-not-encoded-in-its-lie-algebra/|What Information About a Lie Group is Not Encoded in its Lie Algebra?]] | ||
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| <tabbox Abstract> | <tabbox Abstract> | ||
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| + | * [[http://jakobschwichtenberg.com/lie-algebra-able-describe-group/|How is a Lie Algebra able to describe a Group?]] | ||
| + | * [[http://www.wetsavannaanimals.net/wordpress/what-information-about-a-lie-group-is-not-encoded-in-its-lie-algebra/|What Information About a Lie Group is Not Encoded in its Lie Algebra?]] | ||
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| + | ---- | ||
| {{ :advanced_tools:group_theory:adjoint.png?nolink |}} | {{ :advanced_tools:group_theory:adjoint.png?nolink |}} | ||
advanced_tools/group_theory/lie_algebras.1523182031.txt.gz · Last modified: 2018/04/08 10:07 (external edit)
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