Trace form of a Lie algebra representation ($(\cdot ,\cdot {)}_V$)

For a finite-dimensional Lie algebra representation $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$, its trace form is the symmetric bilinear formIt is invariant: $([x,y],z{)}_V=(x,[y,z]{)}_V$.

For a finite-dimensional complex representation of a solvable Lie algebra, the Lie theorem gives a basis in which all representing matrices are upper triangular. Commutators are strictly upper triangular, and their products with upper triangular matrices have zero trace. Thus the derived algebra lies in the radical of a bilinear form. In particular, a nonabelian solvable Lie algebra can carry an invariant bilinear form on a Lie algebra that is nondegenerate without that form coming from a finite-dimensional representation trace.

// Target: semisimple-lie-algebra.bigb

The cubic trace tensor of a Lie algebra representation is cyclic by the cyclic property of the trace. It need not be fully antisymmetric. Its antisymmetric part in the first two arguments is fixed by the Trace form of a Lie algebra representation:

// Target: lie-theory.bigb

Lowering and raising structure constants with the Killing form gives $c_a{}^{mn}{c}_{mn}{}^r=-{\delta }_a{}^r$. Since $c_a{}^{mn}$ is antisymmetric in $m,n$, only the commutator part of the cubic trace tensor of a Lie algebra representation contributes, proving $c_a{}^{mn}{B}_{mnb}=-H_{ab}/2$. In an adapted compact basis with ${\kappa }_{ab}=-{\delta }_{ab}$ and $H_{ab}=-\mu {\delta }_{ab}$, the resulting sign is positive: $+\mu {\delta }_{ab}/2$. Defining the lowered structure constants with a positive Euclidean metric instead would reverse this sign.

// Target: lie-theory.bigb

The trace of a commutator givesIn a basis, cyclicity converts this to $c_{da}{}^{\ell }{B}_{bc\ell }+c_{db}{}^{\ell }{B}_{ca\ell }+c_{dc}{}^{\ell }{B}_{ab\ell }=0$.

// Target: lie-theory.bigb

For a nontrivial finite-dimensional anti-Hermitian Lie algebra representation of a compact Lie algebra that is simple, its Trace form of a Lie algebra representation is a positive multiple of the Killing form. Define $T$ by $H(X,Y)=\kappa (TX,Y)$. Invariance makes $T$ commute with the Adjoint representation of a Lie algebra. Symmetry makes $T$ self-adjoint for $-\kappa$, so its eigenspaces are ideals. Simplicity implies $T=\mu I$. The kernel of the representation is an ideal and hence vanishes; therefore $H(X,X)=-\mathrm{Tr}(d(X{)}^{†}d(X))<0$ for nonzero $X$, proving $\mu >0$.

// Target: lie-theory.bigb