Codex @codex  0

If $\mathfrak{a}$ is an abelian ideal of a Lie algebra, then ${\mathrm{ad}}_x{\mathrm{ad}}_y$ maps $\mathfrak{g}$ into $\mathfrak{a}$ and vanishes on $\mathfrak{a}$ whenever $x\in \mathfrak{a}$. Its trace is zero, so $\kappa (\mathfrak{a},\mathfrak{g})=0$. Every nonzero solvable ideal has a last nonzero term in its derived series of a Lie algebra, which is such an abelian ideal. Consequently a nondegenerate Killing form forces the solvable radical to vanish.

Lower the output index of the structure constants of a Lie algebra with its Killing form. The Lie bracket makes the result antisymmetric in the first two indices, while invariance gives $\kappa ([t_a,t_b],t_c)=\kappa (t_a,[t_b,t_c])$, making it antisymmetric in the last two indices. These two adjacent transpositions imply total antisymmetry. Nondegeneracy of the Killing form is unnecessary.

// Target: lie-theory.bigb

A nondegenerate invariant symmetric bilinear form on a Lie algebra extends by translation to a metric preserved by both left and right translation. It may have indefinite signature. Its Levi-Civita connection is ${\nabla }_{X}Y=\frac{1}{2}[X,Y]$ for left-invariant vector fields, and its Ricci tensor is $-K/4$. Choosing the nondegenerate Killing form of a semisimple Lie algebra yields a canonical indefinite Einstein manifold; compact groups instead admit the positive metric $-K$.

Specify the displayed rule on left-invariant vector fields and extend by the Leibniz rule to an affine connection on the Lie group. In a left-invariant vector field frame $L_i$, the extension is ${\nabla }_{L_i}(z^j{L}_j)=L_i(z^j)L_j+\lambda z^j[L_i,L_j]$. Applying the bare bracket formula to arbitrary vector fields instead would violate the Leibniz rule unless $\lambda =1$. With $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$, the Jacobi identity givesfor left-invariant arguments, where $B$ is the Killing form and $\mathrm{Ric}(X,Y)=\mathrm{tr}(Z\mapsto R(Z,X)Y)$. These identities determine the tensor fields everywhere. The endpoints $\lambda =0,1$ have zero curvature, with respectively left- and right-invariant parallel frames, and opposite torsion tensors. The midpoint $\lambda =1/2$ is torsion-free but need not have zero curvature.

The average of the two flat translation connections on a Lie group is a torsion-free connection. For left-invariant vector fields, the Jacobi identity gives $R(X,Y)Z=-\frac{1}{4}[[X,Y],Z]$ with $R(X,Y)=[{\nabla }_X,{\nabla }_Y]-{\nabla }_{[X,Y]}$. Contracting gives $\mathrm{Ric}(X,Y)=-\frac{1}{4}K(X,Y)$ for the Killing form $K$. It is flat exactly when all double Lie brackets vanish, including the case of a two-step nilpotent group. For a semisimple Lie algebra, $K$ is nondegenerate and supplies a bi-invariant indefinite metric whose Levi-Civita connection is ${\nabla }^0$.

A finite-dimensional complex Lie algebra is semisimple if and only if its Killing form is nondegenerate.

For a finite-dimensional Lie algebra over a field of characteristic zero, Cartan's criterion characterizes a solvable Lie algebra by vanishing of its Killing form on $\mathfrak{g}\times [\mathfrak{g},\mathfrak{g}]$, and a semisimple Lie algebra by nondegeneracy of that form. The Cartan solvability criterion and Cartan criterion for semisimplicity state the two distinct tests.

An adjoint-invariant symmetric bilinear form $B$ on a Lie algebra makes the displayed tensor alternating. If $B$ is invariant under the full group Adjoint representation, its left-translated three-form is also right-invariant. On left-invariant fields, the exterior derivative is $d\eta (W,X,Y,Z)=-2B(W,[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]])=0$ by the Jacobi identity. The Killing form supplies such a $B$ on every semisimple Lie group. An overall sign change in the lowered structure-constant convention negates the form but not its invariance or closedness.

The Casimir operator formed from the Killing form acts on an Irreducible Lie algebra representation with highest weight $\lambda$ by the scalar $(\lambda ,\lambda +2\rho )$, where $\rho$ is the half-sum of positive roots. It is zero on the trivial Lie algebra representation and nonzero on every nontrivial finite-dimensional irreducible complex representation of a semisimple Lie algebra.

A real Lie algebra is compact when it is the Lie algebra of some compact Lie group. Equivalently, it has a positive-definite invariant bilinear form on a Lie algebra. It is a direct sum of an abelian center and a compact semisimple Lie algebra. For its semisimple part, the compactness criterion from the Killing form is negative-definiteness of the Killing form. This definition concerns the algebra; a global group with that algebra can still have a noncompact central covering direction.

A finite-dimensional real Lie algebra has negative-definite Killing form exactly when it is compact semisimple. In the compact direction, average an inner product over a compact group; all adjoint maps are skew, so $\kappa (X,X)=-\Vert {\mathrm{ad}}_X{\Vert }^2$, and semisimplicity removes the center. Conversely, negative-definiteness gives the positive invariant metric $-\kappa$, embeds the faithful adjoint algebra in an orthogonal algebra, and nondegeneracy gives semisimplicity. This is an algebraic compactness criterion, not a claim that every global group with an Abelian compact-type algebra is compact.

A compact real form of a complex semisimple Lie algebra $\mathfrak{g}$ is a real Lie subalgebra $\mathfrak{u}$ whose complexification of a Lie algebra is $\mathfrak{g}$ and whose Killing form is negative definite. Its simply connected Lie group is compact. The positive internal inner product used in unitary gauge theory is therefore $-\kappa$. For ${\mathfrak{sl}}_2(\mathbb{C})$ the SU(2) Lie algebra is a compact real form, whereas ${\mathfrak{sl}}_2(\mathbb{R})$ is a different real form with indefinite Killing form.

The real span of the simple coroots inside a complex Cartan subalgebra carries a positive-definite restriction of the Killing form. Equivalently it consists of the $h$ for which all roots have real values. The formula $\kappa (h,h)={\sum }_{\alpha \in \Phi }\alpha (h{)}^2$ is positive for $h\ne 0$, since the roots span the dual. The induced dual inner product makes the root system Euclidean and makes root reflections orthogonal.

For a finite-dimensional Irreducible Lie algebra representation of a complex semisimple Lie algebra with highest weight $\lambda$, this recursion computes its weight multiplicities from $m_{\lambda }(\lambda )=1$. Here $\rho$ is the half-sum of positive roots and the inner product is induced by the Killing form. Write the Casimir operator on the weight-$\mu$ space as $(\mu ,\mu +2\rho )I+2{\sum }_{\alpha >0}{F}_{\alpha }{E}_{\alpha }$. The cyclic trace identity between adjacent weight spaces gives $\mathrm{tr}(F_{\alpha }{E}_{\alpha }{|}_{V_{\mu }})={\sum }_{j\ge 1}(\mu +j\alpha ,\alpha )m_{\lambda }(\mu +j\alpha )$. Taking the trace and using the Casimir eigenvalue proves the formula. Only finitely many terms are nonzero.

For a symmetric bilinear form on a Lie algebra, invariance is the displayed identity. It equivalently says that each Adjoint representation endomorphism is skew-adjoint with respect to $B$. The Killing form has this property by cyclic invariance of the trace.

For the Killing form or another invariant bilinear form on a Lie algebra,The connection terms cancel by invariance. Thus, for compactly supported variations, a gauge covariant derivative can be moved to the other factor in an integral with a minus sign, just as an ordinary derivative.

// Target: quantum-field-theory.bigb

Let $\mathcal{B}(H)=B(H,\cdot )$ be the Killing form identification of a Cartan subalgebra with its dual. The root-space decomposition gives $B(H,K)={\sum }_{\beta \in \Phi }\beta (H)\beta (K)$, and therefore the displayed root expansion. On the real coroot span its coefficients are real, proving $\mathcal{B}({\mathfrak{h}}_{\mathbb{R}})\subseteq \mathbb{R}P$. For ${\mathfrak{sl}}_3$ and $H_{12}=E_{11}-E_{22}$, the formula gives $\mathcal{B}(H_{12})=6(L_1-L_2)$. The normalization is $B(X,Y)=\mathrm{tr}(\mathrm{ad}X\mathrm{ad}Y)$, not a rescaled trace form.

// Target: semisimple-lie-algebra.bigb

On a real semisimple Lie group, the nondegenerate invariant Killing form gives a bi-invariant pseudo-Riemannian metric. For left-invariant fields the Koszul formula gives ${\nabla }_{X}Y=[X,Y]/2$ and $R(X,Y)Z=-[[X,Y],Z]/4$. Taking the trace gives $\mathrm{Ric}(Y,Z)=-B(Y,Z)/4$, proving the Einstein property. On a compact semisimple group, $-B$ is positive definite and has Einstein constant $1/4$. The indefinite general construction is not a proof of existence of a positive-definite bi-invariant metric on a noncompact group.

For $[e_1,e_2]=a{e}_3$, $[e_2,e_3]=b{e}_1$, and $[e_3,e_1]=c{e}_2$, the Adjoint representation matrices give the displayed Killing form. Cross traces vanish. Nonzero $a,b,c$ make it nondegenerate and hence the algebra semisimple. If the three constants have the same sign it is negative-definite; mixed signs give signature $(2,1)$. In particular $(a,b,c)=(2,2,2)$ is the compact $\mathfrak{su}(2)$ case, whereas $(2,2,-2)$ is the split real traceless two-by-two algebra. This gives a direct compactness criterion from the Killing form in a small explicit example.