Semi-direct sum

The semi-direct sum is a mathematical construction from the theory of Lie algebras .

construction

Let and Lie algebras, be a representation , that is: ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {b}}}$ ${\displaystyle \pi :{\mathfrak {a}}\rightarrow {\rm {End}}({\mathfrak {b}})}$

• ${\displaystyle \pi }$is linear, and for all true . ${\displaystyle a_{1},a_{2}\in {\mathfrak {a}}}$ ${\displaystyle \pi ([a_{1},a_{2}])\,=\,\pi (a_{1})\pi (a_{2})-\pi (a_{2})\pi (a_{1})}$
• ${\displaystyle \pi (a)}$is a derivation for everyone . ${\displaystyle a\in {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {b}}}$

Then there is exactly one bracket on the direct sum of the vector spaces , so that the following applies: ${\displaystyle {\mathfrak {a}}\oplus {\mathfrak {b}}}$ ${\displaystyle [\cdot ,\cdot ]}$

• ${\displaystyle {\mathfrak {a}}\oplus {\mathfrak {b}}}$is with a Lie algebra. ${\displaystyle [\cdot ,\cdot ]}$
• The restriction of the bracket to and corresponds to the brackets given there. ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {b}}}$
• For everyone and applies . ${\displaystyle a\in {\mathfrak {a}}}$ ${\displaystyle b\in {\mathfrak {b}}}$ ${\displaystyle [a,b]\,=\,\pi (a)b}$

Here are and regarded as subspaces of the direct sum. ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {b}}}$

The bracket reads ${\displaystyle {\mathfrak {a}}\oplus {\mathfrak {b}}}$

${\displaystyle [(a_{1},b_{1})\,,\,(a_{2},b_{2})]:=([a_{1},a_{2}]\,,\,[b_{1},b_{2}]+\pi (a_{1})b_{2}-\pi (a_{2})b_{1}),\quad a_{1},a_{2}\in {\mathfrak {a}},\,b_{1},b_{2}\in {\mathfrak {b}}}$.

One calculates that a Lie algebra is given by this definition. This is denoted by and is called the semi-direct sum or the semi-direct product of and . If there can be no misunderstandings with regard to the presentation , leave them out and just write . ${\displaystyle {\mathfrak {a}}\ltimes _{\pi }{\mathfrak {b}}}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {b}}}$ ${\displaystyle \pi }$ ${\displaystyle {\mathfrak {a}}\ltimes {\mathfrak {b}}}$

Remarks

• In the above construction is a Lie subalgebra of the semi-direct sum and even an ideal , that is . ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {b}}}$ ${\displaystyle [{\mathfrak {a}}\ltimes {\mathfrak {b}},{\mathfrak {b}}]\subset {\mathfrak {b}}}$
• If , then we have the direct sum of the Lie algebras. ${\displaystyle \pi =0}$
• Be a Lie algebra over the body and a derivation on it . Then there is a representation and one can educate. This is also called the adjunction of derivation . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle K}$ ${\displaystyle d}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle Kd\subset {\rm {End}}({\mathfrak {g}})}$ ${\displaystyle Kd\ltimes {\mathfrak {g}}}$ ${\displaystyle d}$

Extensions

If and , a short exact sequence of Lie algebras and Lie algebra homomorphisms is obtained ${\displaystyle \iota :{\mathfrak {b}}\rightarrow {\mathfrak {a}}\ltimes {\mathfrak {b}},b\mapsto (0,b)}$ ${\displaystyle q:{\mathfrak {a}}\ltimes {\mathfrak {b}}\rightarrow {\mathfrak {a}},(a,b)\mapsto a}$

${\displaystyle 0\rightarrow {\mathfrak {b}}\,{\xrightarrow {\iota }}\,{\mathfrak {a}}\ltimes {\mathfrak {b}}\,{\xrightarrow {q}}{\mathfrak {a}}\rightarrow 0}$.

Generally one calls short exact sequences

${\displaystyle 0\rightarrow {\mathfrak {b}}\rightarrow {\mathfrak {c}}{\xrightarrow {q}}{\mathfrak {a}}\rightarrow 0}$

or the Lie algebra occurring in it is an extension from to (sometimes you can also find the opposite way of speaking) and such an extension is called decaying if there is a Lie algebra homomorphism with . Accordingly, there is such a disintegrating expansion, because homomorphism does what is desired. ${\displaystyle {\mathfrak {c}}}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {b}}}$ ${\displaystyle \psi :{\mathfrak {a}}\rightarrow {\mathfrak {c}}}$ ${\displaystyle q\circ \psi =\mathrm {id} _{\mathfrak {a}}}$ ${\displaystyle {\mathfrak {a}}\ltimes {\mathfrak {b}}}$ ${\displaystyle {\mathfrak {a}}\rightarrow {\mathfrak {a}}\ltimes {\mathfrak {b}},a\mapsto (a,0)}$

Finally, two extensions and are called equivalent if there is an isomorphism that defines the diagram ${\displaystyle {\mathfrak {c}}_{1}}$ ${\displaystyle {\mathfrak {c}}_{2}}$ ${\displaystyle \varphi :{\mathfrak {c}}_{1}\rightarrow {\mathfrak {c}}_{2}}$

${\displaystyle {\begin{array}{ccccccc}0\rightarrow &{\mathfrak {b}}&\rightarrow &{\mathfrak {c}}_{1}&\rightarrow &{\mathfrak {a}}&\rightarrow 0\\&\parallel &&\downarrow \gamma &&\parallel \\0\rightarrow &{\mathfrak {b}}&\rightarrow &{\mathfrak {c}}_{2}&\rightarrow &{\mathfrak {a}}&\rightarrow 0\\\end{array}}}$

makes commutative. With the help of the semi-direct sum one can characterize decaying extensions as follows:

An extension

${\displaystyle 0\rightarrow {\mathfrak {b}}\rightarrow {\mathfrak {c}}\rightarrow {\mathfrak {a}}\rightarrow 0}$

von Lie algebras are decaying if and only if they are equivalent to the semi-direct sum

${\displaystyle 0\rightarrow {\mathfrak {b}}\,{\xrightarrow {\iota }}\,{\mathfrak {a}}\ltimes {\mathfrak {b}}\,{\xrightarrow {q}}{\mathfrak {a}}\rightarrow 0}$

is.

See also

• Affine Lie algebra
• Virasoro Algebra

Individual evidence

1. ^ Anthony W. Knapp: Lie Groups Beyond an Introduction . Birkhäuser, 2002, ISBN 0817642595 , chap. I.4: Semidirect products of Lie algebras
2. ↑ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , II.1.13
3. ^ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , II.4.4