In mathematics, the Killing form, called afterwards Wilhelm Killing, is a symmetric bilinear anatomy that plays a basal role in the theories of Lie groups and Lie algebras. The Killing anatomy was about alien into Lie algebra approach by Élie Cartan (1894) in his thesis; although Killing had ahead fabricated a casual acknowledgment of it he fabricated no austere use of it.
Saturday, 13 August 2011
Various definitions
Consider a Lie algebra g over a acreage K. Every aspect x of g defines the adjoint endomorphism ad(x) (also accounting as adx) of g with the advice of the Lie bracket, as
\textrm{ad}(x)(y) = [x, y].\
Now, admitting g is of bound dimension, the trace of the agreement of two such endomorphisms defines a symmetric bilinear form
B(x, y) = trace(ad(x)ad(y)),
with ethics in K, the Killing anatomy on g.
\textrm{ad}(x)(y) = [x, y].\
Now, admitting g is of bound dimension, the trace of the agreement of two such endomorphisms defines a symmetric bilinear form
B(x, y) = trace(ad(x)ad(y)),
with ethics in K, the Killing anatomy on g.
Properties
The Killing anatomy B is bilinear and symmetric.
The Killing anatomy is an invariant form, in the faculty that it has the 'associativity' property
B([x,y],z)=B(x,[y,z]),
area [,] is the Lie bracket.
If g is a simple Lie algebra again any invariant symmetric bilinear anatomy on g is a scalar assorted of the Killing form.
The Killing anatomy is additionally invariant beneath automorphisms s of the algebra g, that is,
B(s(x),s(y)) = B(x,y)
for s in Aut(g).
The Cartan archetype states that a Lie algebra is semisimple if and alone if the Killing anatomy is non-degenerate.
The Killing anatomy of a nilpotent Lie algebra is analogously zero.
If I and J are two ethics in a Lie algebra g with aught intersection, again I and J are erect subspaces with account to the Killing form.
If a accustomed Lie algebra g is a absolute sum of its ethics I1,...,In, again the Killing anatomy of g is the absolute sum of the Killing forms of the alone summands.
The Killing anatomy is an invariant form, in the faculty that it has the 'associativity' property
B([x,y],z)=B(x,[y,z]),
area [,] is the Lie bracket.
If g is a simple Lie algebra again any invariant symmetric bilinear anatomy on g is a scalar assorted of the Killing form.
The Killing anatomy is additionally invariant beneath automorphisms s of the algebra g, that is,
B(s(x),s(y)) = B(x,y)
for s in Aut(g).
The Cartan archetype states that a Lie algebra is semisimple if and alone if the Killing anatomy is non-degenerate.
The Killing anatomy of a nilpotent Lie algebra is analogously zero.
If I and J are two ethics in a Lie algebra g with aught intersection, again I and J are erect subspaces with account to the Killing form.
If a accustomed Lie algebra g is a absolute sum of its ethics I1,...,In, again the Killing anatomy of g is the absolute sum of the Killing forms of the alone summands.
Matrix elements
Given a basis ei of the Lie algebra g, the matrix elements of the Killing form are given by
B^{ij}= tr (\textrm{ad}(e^i)\circ \textrm{ad}(e^j)) / I_{ad}
where Iad is the Dynkin index of the adjoint representation of g.
Here
\left(\textrm{ad}(e^i) \circ \textrm{ad}(e^j)\right)(e^k)= [e^i, [e^j, e^k]] = {c^{im}}_{n} {c^{jk}}_{m} e^n
in Einstein summation notation and so we can write
B^{ij} = \frac{1}{I_{\textrm{ad}}} {c^{im}}_{n} {c^{jn}}_{m}
where the {c^{ij}}_{k} are the structure constants of the Lie algebra. The Killing form is the simplest 2-tensor that can be formed from the structure constants.
In the above indexed definition, we are careful to distinguish upper and lower indexes (co- and contra-variant indexes). This is because, in many cases, the Killing form can be used as a metric tensor on a manifold, in which case the distinction becomes an important one for the transformation properties of tensors. When the Lie algebra is semisimple, its Killing form is nondegenerate, and hence can be used as a metric tensor to raise and lower indexes. In this case, it is always possible to choose a basis for g such that the structure constants with all upper indexes are completely antisymmetric.
B^{ij}= tr (\textrm{ad}(e^i)\circ \textrm{ad}(e^j)) / I_{ad}
where Iad is the Dynkin index of the adjoint representation of g.
Here
\left(\textrm{ad}(e^i) \circ \textrm{ad}(e^j)\right)(e^k)= [e^i, [e^j, e^k]] = {c^{im}}_{n} {c^{jk}}_{m} e^n
in Einstein summation notation and so we can write
B^{ij} = \frac{1}{I_{\textrm{ad}}} {c^{im}}_{n} {c^{jn}}_{m}
where the {c^{ij}}_{k} are the structure constants of the Lie algebra. The Killing form is the simplest 2-tensor that can be formed from the structure constants.
In the above indexed definition, we are careful to distinguish upper and lower indexes (co- and contra-variant indexes). This is because, in many cases, the Killing form can be used as a metric tensor on a manifold, in which case the distinction becomes an important one for the transformation properties of tensors. When the Lie algebra is semisimple, its Killing form is nondegenerate, and hence can be used as a metric tensor to raise and lower indexes. In this case, it is always possible to choose a basis for g such that the structure constants with all upper indexes are completely antisymmetric.
Connection with real forms
Suppose that g is a semisimple Lie algebra over the acreage of absolute numbers. By Cartan's criterion, the Killing anatomy is nondegenerate, and can be diagonalized in a acceptable base with the askew entries +1 or -1. By Sylvester's law of inertia, the cardinal of absolute entries is an invariant of the bilinear form, i.e. it does not depend on the best of the diagonalizing basis, and is alleged the basis of the Lie algebra g. This is a cardinal amid 0 and the ambit of g which is an important invariant of the absolute Lie algebra. In particular, a absolute Lie algebra g is alleged bunched if the Killing anatomy is abrogating definite. It is accepted that beneath the Lie correspondence, bunched Lie algebras accord to bunched Lie groups.
If gC is a semisimple Lie algebra over the circuitous numbers, again there are several non-isomorphic absolute Lie algebras whose complexification is gC, which are alleged its absolute forms. It turns out that every circuitous semisimple Lie algebra admits a different (up to isomorphism) bunched absolute anatomy g. The absolute forms of a accustomed circuitous semisimple Lie algebra are frequently labeled by the absolute basis of apathy of their Killing form.
For example, the circuitous appropriate beeline algebra sl(2,C) has two absolute forms, the absolute appropriate beeline algebra, denoted sl(2,R), and the appropriate unitary algebra, denoted su(2). The aboriginal one is noncompact, the alleged breach absolute form, and its Killing anatomy has signature (2,1). The additional one is the bunched absolute anatomy and its Killing anatomy is abrogating definite, i.e. has signature (0,3). The agnate Lie groups are the noncompact accumulation SL(2,R) of 2 by 2 absolute matrices with the assemblage account and the appropriate unitary accumulation SU(2), which is compact.
If gC is a semisimple Lie algebra over the circuitous numbers, again there are several non-isomorphic absolute Lie algebras whose complexification is gC, which are alleged its absolute forms. It turns out that every circuitous semisimple Lie algebra admits a different (up to isomorphism) bunched absolute anatomy g. The absolute forms of a accustomed circuitous semisimple Lie algebra are frequently labeled by the absolute basis of apathy of their Killing form.
For example, the circuitous appropriate beeline algebra sl(2,C) has two absolute forms, the absolute appropriate beeline algebra, denoted sl(2,R), and the appropriate unitary algebra, denoted su(2). The aboriginal one is noncompact, the alleged breach absolute form, and its Killing anatomy has signature (2,1). The additional one is the bunched absolute anatomy and its Killing anatomy is abrogating definite, i.e. has signature (0,3). The agnate Lie groups are the noncompact accumulation SL(2,R) of 2 by 2 absolute matrices with the assemblage account and the appropriate unitary accumulation SU(2), which is compact.
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