Killing form - Revision history

Vipul: /* Other facts */

2009-07-18T13:09:19Z

Other facts

← Older revision		Revision as of 13:09, 18 July 2009
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	* [[Killing form on subalgebra not equals restriction of Killing form]]		* [[Killing form on subalgebra not equals restriction of Killing form]]
	* [[Abelian ideal is degenerate for Killing form]]		* [[Abelian ideal is degenerate for Killing form]]
			* [[Nilpotent ideal is degenerate for Killing form]]

Vipul at 13:05, 18 July 2009

2009-07-18T13:05:42Z

← Older revision		Revision as of 13:05, 18 July 2009
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	==Definition==		==Definition==

	Suppose <math>L</math> is a finite-dimensional [[Lie algebra]] over a [[field]] <math>k</math>. The '''Killing form''' on <math>L</math> is a <math>k</math>-bilinear form <math>\kappa</math> on <math>L</math> defined as follows:		Suppose <math>L</math> is a finite-dimensional [[Lie algebra]] over a [[field]] <math>F</math>. The '''Killing form''' on <math>L</math> is a <math>F</math>-bilinear form <math>\kappa</math> on <math>L</math> defined as follows:

	<math>\kappa(x,y) = \operatorname{tr}(\operatorname{ad}(x) \circ \operatorname{ad}(y))</math>.		<math>\kappa(x,y) = \operatorname{tr}(\operatorname{ad}(x) \circ \operatorname{ad}(y))</math>.

	Here, <math>\operatorname{ad}(x)</math> is the adjoint action of <math>x</math> on <math>L</math> (see [[Lie ring acts as derivations by adjoint action]]), viewed as a <math>k</math>-linear map from <math>L</math> to itself. The composite is thus also a <math>k</math>-linear map from <math>L</math> to itself. <math>\operatorname{tr}</math> computes the trace of this linear map.		Here, <math>\operatorname{ad}(x)</math> is the adjoint action of <math>x</math> on <math>L</math> (see [[Lie ring acts as derivations by adjoint action]]), viewed as a <math>F</math>-linear map from <math>L</math> to itself. The composite is thus also a <math>F</math>-linear map from <math>L</math> to itself. <math>\operatorname{tr}</math> computes the trace of this linear map.

	Note that the Killing form can be defined and makes sense only for [[Lie algebra]]s and not for the more general [[Lie ring]]s.		Note that the Killing form can be defined and makes sense only for [[Lie algebra]]s and not for the more general [[Lie ring]]s.
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	Cartan's criteria rely on [[Lie's theorem]], which in turn depends on the field being algebraically closed and of characteristic zero.		Cartan's criteria rely on [[Lie's theorem]], which in turn depends on the field being algebraically closed and of characteristic zero.

	* [[Cartan's first criterion]]: This states that if <math>k</math> is an algebraically closed field of characteristic zero (e.g., the field of complex numbers), then a Lie algebra <math>L</math> over <math>k</math> is solvable if and only if <math>\kappa(x,y) = 0</math> for all <math>x \in L</math> and <math>y \in [L,L]</math>.		* [[Cartan's first criterion]]: This states that if <math>F</math> is an algebraically closed field of characteristic zero (e.g., the field of complex numbers), then a Lie algebra <math>L</math> over <math>F</math> is solvable if and only if <math>\kappa(x,y) = 0</math> for all <math>x \in L</math> and <math>y \in [L,L]</math>.
	* [[Cartan's second criterion]]: This states that if <math>k</math> is an algebraically closed field of characteristic zero (e.g., the field of complex numbers), then a Lie algebra over <math>k</math> is [[semisimple Lie algebra\|semisimple]] if and only if the Killing form is nondegenerate on the algebra.		* [[Cartan's second criterion]]: This states that if <math>F</math> is an algebraically closed field of characteristic zero (e.g., the field of complex numbers), then a Lie algebra over <math>F</math> is [[semisimple Lie algebra\|semisimple]] if and only if the Killing form is nondegenerate on the algebra.

	===Other facts===		===Other facts===

Vipul at 00:05, 18 July 2009

2009-07-18T00:05:10Z

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 ==Facts==
 ===Cartan's criteria===
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 * [[Cartan's first criterion]]: This states that if <math>k</math> is an algebraically closed field of characteristic zero (e.g., the field of complex numbers), then a Lie algebra <math>L</math> over <math>k</math> is solvable if and only if <math>\kappa(x,y) = 0</math> for all <math>x \in L</math> and <math>y \in [L,L]</math>.
 * [[Cartan's second criterion]]: This states that if <math>k</math> is an algebraically closed field of characteristic zero (e.g., the field of complex numbers), then a Lie algebra over <math>k</math> is [[semisimple Lie algebra|semisimple]] if and only if the Killing form is nondegenerate on the algebra.

Vipul at 15:51, 17 July 2009

2009-07-17T15:51:59Z

← Older revision		Revision as of 15:51, 17 July 2009
Line 1:		Line 1:
	==Definition==		==Definition==

	Suppose <math>L</math> is a [[Lie algebra]] over a [[field]] <math>k</math>. The '''Killing form''' on <math>L</math> is a <math>k</math>-bilinear form <math>\kappa</math> on <math>L</math> defined as follows:		Suppose <math>L</math> is a finite-dimensional [[Lie algebra]] over a [[field]] <math>k</math>. The '''Killing form''' on <math>L</math> is a <math>k</math>-bilinear form <math>\kappa</math> on <math>L</math> defined as follows:

	<math>\kappa(x,y) = \operatorname{tr}(\operatorname{ad}(x) \circ \operatorname{ad}(y))</math>.		<math>\kappa(x,y) = \operatorname{tr}(\operatorname{ad}(x) \circ \operatorname{ad}(y))</math>.

Vipul: Created page with '==Definition== Suppose $L$ is a Lie algebra over a field $k$ . The '''Killing form''' on $L$ is a $k$ -bilinear form $\kap…'$

2009-07-16T16:24:20Z

Created page with '==Definition== Suppose <math>L</math> is a Lie algebra over a field <math>k</math>. The '''Killing form''' on <math>L</math> is a <math>k</math>-bilinear form <math>\kap…'

New page

==Definition==

Suppose <math>L</math> is a [[Lie algebra]] over a [[field]] <math>k</math>. The '''Killing form''' on <math>L</math> is a <math>k</math>-bilinear form <math>\kappa</math> on <math>L</math> defined as follows:

<math>\kappa(x,y) = \operatorname{tr}(\operatorname{ad}(x) \circ \operatorname{ad}(y))</math>.

Here, <math>\operatorname{ad}(x)</math> is the adjoint action of <math>x</math> on <math>L</math> (see [[Lie ring acts as derivations by adjoint action]]), viewed as a <math>k</math>-linear map from <math>L</math> to itself. The composite is thus also a <math>k</math>-linear map from <math>L</math> to itself. <math>\operatorname{tr}</math> computes the trace of this linear map.

Note that the Killing form can be defined and makes sense only for [[Lie algebra]]s and not for the more general [[Lie ring]]s.

==Facts==

===Cartan's criteria===

Cartan's criteria rely on [[Lie's theorem]], which in turn depends on the field being algebraically closed and of characteristic zero.

* [[Cartan's first criterion]]: This states that if <math>k</math> is an algebraically closed field of characteristic zero (e.g., the field of complex numbers), then a Lie algebra <math>L</math> over <math>k</math> is solvable if and only if <math>\kappa(x,y) = 0</math> for all <math>x \in L</math> and <math>y \in [L,L]</math>.
* [[Cartan's second criterion]]: This states that if <math>k</math> is an algebraically closed field of characteristic zero (e.g., the field of complex numbers), then a Lie algebra over <math>k</math> is [[semisimple Lie algebra|semisimple]] if and only if the Killing form is nondegenerate on the algebra.

Killing form - Revision history

Vipul: /* Other facts */

Vipul at 13:05, 18 July 2009

Vipul at 00:05, 18 July 2009

Vipul at 15:51, 17 July 2009

Vipul: Created page with '==Definition== Suppose L is a Lie algebra over a field k. The '''Killing form''' on L is a k-bilinear form \kap…'

Vipul: Created page with '==Definition== Suppose $L$ is a Lie algebra over a field $k$ . The '''Killing form''' on $L$ is a $k$ -bilinear form $\kap…'$