Killing form

Last updated November 29, 2024

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras.^[1]

History and name

The Killing form was essentially introduced into Lie algebra theory by ÉlieCartan ( 1894 ) in his thesis. In a historical survey of Lie theory, Borel (2001) has described how the term "Killing form" first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached.^[2] Some other authors now employ the term "Cartan-Killing form".^{[ citation needed ]} At the end of the 19th century, Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which it follows that the Killing form (i.e. the degree 2 coefficient) is invariant, but he did not make much use of the fact. A basic result that Cartan made use of was Cartan's criterion, which states that the Killing form is non-degenerate if and only if the Lie algebra is a direct sum of simple Lie algebras.^[2]

Definition

Consider a Lie algebra ${\mathfrak {g}}$ over a field $K$ . Every element $x$ of ${\mathfrak {g}}$ defines the adjoint endomorphism $ad(x)$ (also written as $ad x$ ) of ${\mathfrak {g}}$ with the help of the Lie bracket, as

\operatorname {ad} (x)(y)=[x,y].

Now, supposing ${\mathfrak {g}}$ is of finite dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form

B(x,y)=\operatorname {trace} (\operatorname {ad} (x)\circ \operatorname {ad} (y)),

with values in $K$ , the Killing form on ${\mathfrak {g}}$ .

Properties

The following properties follow as theorems from the above definition.

The Killing form $B$ is bilinear and symmetric.
The Killing form is an invariant form, as are all other forms obtained from Casimir operators. The derivation of Casimir operators vanishes; for the Killing form, this vanishing can be written as

B([x,y],z)=B(x,[y,z])

where [ , ] is the Lie bracket.

If ${\mathfrak {g}}$ is a simple Lie algebra then any invariant symmetric bilinear form on ${\mathfrak {g}}$ is a scalar multiple of the Killing form.
The Killing form is also invariant under automorphisms $s$ of the algebra ${\mathfrak {g}}$ , that is,

B(s(x),s(y))=B(x,y)

for

s

in

{\mathfrak {g}}

.

The Cartan criterion states that a Lie algebra is semisimple if and only if the Killing form is non-degenerate.
The Killing form of a nilpotent Lie algebra is identically zero.
If $I, J$ are two ideals in a Lie algebra ${\mathfrak {g}}$ with zero intersection, then $I$ and $J$ are orthogonal subspaces with respect to the Killing form.
The orthogonal complement with respect to $B$ of an ideal is again an ideal.^[3]
If a given Lie algebra ${\mathfrak {g}}$ is a direct sum of its ideals $I 1,..., I n$ , then the Killing form of ${\mathfrak {g}}$ is the direct sum of the Killing forms of the individual summands.

Matrix elements

Given a basis $e i$ of the Lie algebra ${\mathfrak {g}}$ , the matrix elements of the Killing form are given by

B_{ij}=\mathrm {trace} (\mathrm {ad} (e_{i})\circ \mathrm {ad} (e_{j})).

Here

\left({\textrm {ad}}(e_{i})\circ {\textrm {ad}}(e_{j})\right)(e_{k})=[e_{i},[e_{j},e_{k}]]=[e_{i},{c_{jk}}^{m}e_{m}]={c_{im}}^{n}{c_{jk}}^{m}e_{n}

in Einstein summation notation, where the $c ij k$ are the structure coefficients of the Lie algebra. The index $k$ functions as column index and the index $n$ as row index in the matrix $ad(e i)ad(e j)$ . Taking the trace amounts to putting $k = n$ and summing, and so we can write

B_{ij}={c_{im}}^{n}{c_{jn}}^{m}

The Killing form is the simplest 2-tensor that can be formed from the structure constants. The form itself is then $B=B_{ij}e^{i}\otimes e^{j}.$

In the above indexed definition, we are careful to distinguish upper and lower indices (co- and contra-variant indices). 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 over a zero-characteristic field, 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 ${\mathfrak {g}}$ such that the structure constants with all upper indices are completely antisymmetric.

The Killing form for some Lie algebras ${\mathfrak {g}}$ are (for $X, Y$ in ${\mathfrak {g}}$ viewed in their fundamental matrix representation):^{[ citation needed ]}

${\mathfrak {g}}$	$B(X,Y)$	Classification	Dual coxeter number
${\mathfrak {gl}}(n,\mathbb {R} )$	$2n{\text{tr}}(XY)-2{\text{tr}}(X){\text{tr}}(Y)$	-	-
${\mathfrak {sl}}(n,\mathbb {R} ),n\geq 2$	$2n{\text{tr}}(XY)$	$A_{n-1}$	$n$
${\mathfrak {su}}(n),n\geq 2$	$2n{\text{tr}}(XY)$	$A_{n-1}$	$n$
${\mathfrak {so}}(n),n\geq 2$	$(n-2){\text{tr}}(XY)$	$B_{m},n=2m+1$ for $n$ odd. $D_{m},n=2m$ for $n$ even.	$n-2$
${\mathfrak {so}}(n,\mathbb {C} ),n\geq 2$	$(n-2){\text{tr}}(XY)$	$B_{m},n=2m+1$ for $n$ odd. $D_{m},n=2m$ for $n$ even.	$n-2$
${\mathfrak {sp}}(2n,\mathbb {R} ),n\geq 1$	$2(n+1){\text{tr}}(XY)$	$C_{n}$	$n+1$
${\mathfrak {sp}}(n),n\geq 1$	$2(n+1){\text{tr}}(XY)$	$C_{n}$	$n+1$

The table shows that the Dynkin index for the adjoint representation is equal to twice the dual Coxeter number.

Connection with real forms

Suppose that ${\mathfrak {g}}$ is a semisimple Lie algebra over the field of real numbers $\mathbb {R}$ . By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries $\pm1$ . By Sylvester's law of inertia, the number of positive entries is an invariant of the bilinear form, i.e. it does not depend on the choice of the diagonalizing basis, and is called the index of the Lie algebra ${\mathfrak {g}}$ . This is a number between $0$ and the dimension of ${\mathfrak {g}}$ which is an important invariant of the real Lie algebra. In particular, a real Lie algebra ${\mathfrak {g}}$ is called compact if the Killing form is negative definite (or negative semidefinite if the Lie algebra is not semisimple). Note that this is one of two inequivalent definitions commonly used for compactness of a Lie algebra; the other states that a Lie algebra is compact if it corresponds to a compact Lie group. The definition of compactness in terms of negative definiteness of the Killing form is more restrictive, since using this definition it can be shown that under the Lie correspondence, compact Lie algebras correspond to compact semisimple Lie groups.

If ${\mathfrak {g}}_{\mathbb {C} }$ is a semisimple Lie algebra over the complex numbers, then there are several non-isomorphic real Lie algebras whose complexification is ${\mathfrak {g}}_{\mathbb {C} }$ , which are called its real forms. It turns out that every complex semisimple Lie algebra admits a unique (up to isomorphism) compact real form ${\mathfrak {g}}$ . The real forms of a given complex semisimple Lie algebra are frequently labeled by the positive index of inertia of their Killing form.

For example, the complex special linear algebra ${\mathfrak {sl}}(2,\mathbb {C} )$ has two real forms, the real special linear algebra, denoted ${\mathfrak {sl}}(2,\mathbb {R} )$ , and the special unitary algebra, denoted ${\mathfrak {su}}(2)$ . The first one is noncompact, the so-called split real form, and its Killing form has signature $(2, 1)$ . The second one is the compact real form and its Killing form is negative definite, i.e. has signature $(0, 3)$ . The corresponding Lie groups are the noncompact group $\mathrm {SL} (2,\mathbb {R} )$ of $2 \times 2$ real matrices with the unit determinant and the special unitary group $\mathrm {SU} (2)$ , which is compact.

Trace forms

Let ${\mathfrak {g}}$ be a finite-dimensional Lie algebra over the field $K$ , and ${\displaystyle \rho$ be a Lie algebra representation. Let ${\text{Tr}}_{V}:{\text{End}}(V)\rightarrow K$ be the trace functional on $V$ . Then we can define the trace form for the representation $\rho$ as

{\text{Tr}}_{\rho }:{\mathfrak {g}}\times {\mathfrak {g}}\rightarrow K,

{\text{Tr}}_{\rho }(X,Y)={\text{Tr}}_{V}(\rho (X)\rho (Y)).

Then the Killing form is the special case that the representation is the adjoint representation, ${\text{Tr}}_{\text{ad}}=B$ .

It is easy to show that this is symmetric, bilinear and invariant for any representation $\rho$ .

If furthermore ${\mathfrak {g}}$ is simple and $\rho$ is irreducible, then it can be shown ${\text{Tr}}_{\rho }=I(\rho )B$ where $I(\rho )$ is the index of the representation.

Citations

↑ Kirillov 2008, p. 102.
1 2 Borel 2001 , p. 5
↑ Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. See page 207.

Related Research Articles

<span class="mw-page-title-main">Lie algebra representation</span> Writing Lie algebra sets as matrices

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space $together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.$

In mathematics, a Casimir element is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group.

In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles. It is built on the earlier Borel–Weil theorem of Armand Borel and André Weil, dealing just with the space of sections, the extension to higher cohomology groups being provided by Raoul Bott. One can equivalently, through Serre's GAGA, view this as a result in complex algebraic geometry in the Zariski topology.

In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.

In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.

<span class="mw-page-title-main">Cartan subalgebra</span> Nilpotent subalgebra of a Lie algebra

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra $of a Lie algebra that is self-normalising. They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic .$

<span class="mw-page-title-main">Semisimple Lie algebra</span> Direct sum of simple Lie algebras

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras.

In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.

In mathematics, a maximal compact subgroupK of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups.

In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edward Witten. A WZW model is associated to a Lie group, and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra. By extension, the name WZW model is sometimes used for any conformal field theory whose symmetry algebra is an affine Lie algebra.

<span class="mw-page-title-main">Symmetric space</span> (pseudo-)Riemannian manifold whose geodesics are reversible

In mathematics, a symmetric space is a Riemannian manifold whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.

In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on $defined by the formula$

In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by Hermann Weyl. There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula.

In mathematics, the Dynkin index $of finite-dimensional highest-weight representations of a compact simple Lie algebra relates their trace forms via$

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L²(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L¹ functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations.

<span class="mw-page-title-main">Real form (Lie theory)</span>

In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g₀ is called a real form of a complex Lie algebra g if g is the complexification of g₀:

<span class="mw-page-title-main">Quadratic Lie algebra</span>

A quadratic Lie algebra is a Lie algebra together with a compatible symmetric bilinear form. Compatibility means that it is invariant under the adjoint representation. Examples of such are semisimple Lie algebras, such as su(n) and sl(n,R).

<span class="mw-page-title-main">Glossary of Lie groups and Lie algebras</span>

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.

<span class="mw-page-title-main">Representation theory of semisimple Lie algebras</span>

In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra ; in particular, it gives a way to parametrize irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight.

References

Borel, Armand (2001), Essays in the history of Lie groups and algebraic groups, History of Mathematics, vol. 21, American Mathematical Society and the London Mathematical Society, ISBN 0821802887
Bump, Daniel (2004), Lie Groups, Graduate Texts In Mathematics, vol. 225, Springer, doi:10.1007/978-1-4614-8024-2, ISBN 978-0-387-21154-1
Cartan, Élie (1894), Sur la structure des groupes de transformations finis et continus, Thesis, Nony
Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN 0-521-48412-X
Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
"Killing form", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
Kirillov, Alexander Jr. (2008), An introduction to Lie groups and Lie algebras, Cambridge Studies in Advanced Mathematics, vol. 113, Cambridge University Press, CiteSeerX 10.1.1.173.1452 , doi:10.1017/CBO9780511755156, ISBN 978-0-521-88969-8, MR 2440737

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[FOOTNOTEKirillov2008102-1] Kirillov 2008, p. 102.

[Borelp5-2] 1 2 Borel 2001 , p. 5

[3] Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. See page 207.

[1]

[2]

[3]