# Book: An

Understand {$A_n$}.

The Dynkin diagram describes the commutator of the Lie algebra. The nodes in the diagram are the dimensions of the commutator. Nodes which are not linked are independent and commute with each other.

The plainest Dynkin diagram is the chain {$A_n$} that determines (as a compact Lie group) the unitary group. The "two out of three property" implies that the unitary structure can be understood simultaneously as an orthogonal structure, a complex structure and a symplectic structure.

{$\mathrm{U}(n)=\mathrm{O}(2n)\cap\mathrm{GL}(n,\mathbb{C})\cap\mathrm{Sp}(2n,\mathbb{R})$}

Wikipedia: a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be compatible (meaning that one uses the same {$J$} in the complex structure and the symplectic form, and that this {$J$} is orthogonal; writing all the groups as matrix groups fixes a {$J$} (which is orthogonal) and ensures compatibility).

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Puslapis paskutinį kartą pakeistas 2019 rugsėjo 06 d., 17:36