mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Killing form, named after

Wilhelm Killing Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of Mü ...

, is a

symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinea ...

that plays a basic role in the theories of

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

s and

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

s. 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.

History and name

The Killing form was essentially introduced into Lie algebra theory by in his thesis. In a historical survey of Lie theory, 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 A misnomer is a name that is incorrectly or unsuitably applied. Misnomers often arise because something was named long before its correct nature was known, or because an earlier form of something has been replaced by a later form to which the name ...

, since the form had previously been used by Lie theorists, without a name attached. Some other authors now employ the term ''" Cartan-Killing form"''. 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 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 f ...

, which states that the Killing form is non-degenerate if and only if the Lie algebra is a direct sum of simple Lie algebras.

Definition

Consider a

\mathfrak g

over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

. Every element of

\mathfrak g

defines the

adjoint endomorphism In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...

(also written as ) of

\mathfrak g

with the help of the Lie bracket, as :

\operatorname(x)(y) =

, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...

Now, supposing

\mathfrak g

is of finite dimension, the

trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...

of the composition of two such endomorphisms defines a

B(x, y) = \operatorname(\operatorname(x) \circ \operatorname(y)),

with values in , the Killing form on

\mathfrak g

Properties

The following properties follow as theorems from the above definition. * The Killing form is bilinear and symmetric. * The Killing form is an invariant form, as are all other forms obtained from

Casimir operator In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...

s. The

derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...

of Casimir operators vanishes; for the Killing form, this vanishing can be written as ::

B(

z) = B(x, , z : where , is the

Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

. * If

\mathfrak g

is a

simple Lie algebra In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of s ...

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 of the algebra

\mathfrak g

, that is, ::

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

:for in

\mathfrak g

. * The

Cartan criterion 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 bilinea ...

states that a Lie algebra is

semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

if and only if the Killing form is

non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent defin ...

. * The Killing form of a

nilpotent Lie algebra In mathematics, a Lie algebra \mathfrak is nilpotent if its lower central series terminates in the zero subalgebra. The ''lower central series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq mathfrak, ideals in a Lie algebra

\mathfrak g

with zero intersection, then and are orthogonal subspaces with respect to the Killing form. * The orthogonal complement with respect to of an ideal is again an ideal. See page 207. * If a given Lie algebra

\mathfrak g

is a direct sum of its ideals , then the Killing form of

\mathfrak g

is the direct sum of the Killing forms of the individual summands.

Matrix elements

Given a basis of the Lie algebra

\mathfrak g

, the matrix elements of the Killing form are given by :

B_= \mathrm(\mathrm(e_i) \circ \mathrm(e_j)).

Here :

\left(\textrm(e_i) \circ \textrm(e_j)\right)(e_k)=_i,_[e_j,_e_k_=_[e_i,_^e_m.html" ;"title="_j,_e_k.html" ;"title="_i, [e_j, e_k">_i, [e_j, e_k = [e_i, ^e_m">_j,_e_k.html" ;"title="_i, [e_j, e_k">_i, [e_j, e_k = [e_i, ^e_m= ^ ^ e_n

in Einstein summation notation, where the are the Structure constants, structure coefficients of the Lie algebra. The index functions as column index and the index as row index in the matrix . Taking the trace amounts to putting and summing, and so we can write :

B_ = ^ ^

The Killing form is the simplest 2-

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

that can be formed from the structure constants. The form itself is then

B=B_ 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

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 in

\mathfrak g

viewed in their fundamental matrix representation):

Connection with real forms

Suppose that

\mathfrak g

is a

semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra i ...

over the field of real numbers

\mathbb R

. By

, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries . By

Sylvester's law of inertia Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if ''A'' is the symmetric matrix that defines the quadra ...

, 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 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 In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: * Negative-definite bilinear form * Negative-definite quadratic form * Nega ...

(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 A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be int ...

compact Lie algebra In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebr ...

s correspond to

compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...

s. If

\mathfrak g_

is a semisimple Lie algebra over the complex numbers, then there are several non-isomorphic real Lie algebras whose

complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include t ...

\mathfrak g_

, 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 (2, \mathbb C)

has two real forms, the real special linear algebra, denoted

\mathfrak (2, \mathbb R)

, and the special unitary algebra, denoted

\mathfrak (2)

. The first one is noncompact, the so-called split real form, and its Killing form has signature . The second one is the compact real form and its Killing form is negative definite, i.e. has signature . The corresponding Lie groups are the noncompact group

\mathrm (2, \mathbb R)

of real matrices with the unit determinant and the special unitary group

\mathrm (2)

, which is compact.

Trace forms

Let

\mathfrak

be a finite-dimensional Lie algebra over the field

K

, and

\rho:\mathfrak\rightarrow \text(V)

be a Lie algebra representation. Let

\text_:\text(V)\rightarrow K

be the trace functional on

V

. Then we can define the trace form for the representation

\rho

as :

\text_\rho:\mathfrak\times\mathfrak\rightarrow K,

\text_\rho(X,Y) = \text_V(\rho(X)\rho(Y)).

Then the Killing form is the special case that the representation is the adjoint representation,

\text_\text = B

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

\rho

. If furthermore

\mathfrak

is simple and

\rho

is irreducible, then it can be shown

\text_\rho = I(\rho)B

where

I(\rho)

is the index of the representation.

Citations

References

* * * * * * * {{refend Lie groups Lie algebras