Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 3 / 3

Preserving the symmetric bilinear form infinitesimally means $Z^{T}M+MZ=0$. In blocks of sizes $2,2,1$, write $Z$ as a general block matrix and multiply by the given $M$. The condition says that $MZ$ is skew-symmetric. It forces the lower-right entry to be zero, the middle diagonal block to be the negative transpose of the upper-left block, and the two off-diagonal two-by-two blocks to be skew-symmetric. Thus the general element of the Special orthogonal Lie algebra iswhere $A$ is an arbitrary two-by-two matrix and $p,q$ are arbitrary two-component columns. More explicitly, using ten independent complex parameters,Conversely, substituting this matrix into $Z^{T}M+MZ$ gives zero, so there are no further restrictions. In particular $\dim {\mathfrak{so}}_5=4+1+1+2+2=10$.

The diagonal Cartan subalgebra isPut ${\epsilon }_1=L_1$ and ${\epsilon }_2=L_2$; then $L_3=-{\epsilon }_1$, $L_4=-{\epsilon }_2$ and $L_5=0$ on $\mathfrak{h}$. For the matrix unit $E_{ij}$, $[H,E_{ij}]=(L_i-L_j)(H)E_{ij}$. Applying this to the general matrix in part (i) gives the following one-dimensional root spaces and chosen root vectors:Each row satisfies $[H,X_{\alpha }]=\alpha (H)X_{\alpha }$ and belongs to the Special orthogonal Lie algebra. The two diagonal parameters and eight root vectors span the ten-parameter matrix from part (i), giving the explicit root-space decompositionThis is the B2 root system, with short roots on the coordinate axes and long roots on the diagonals.

For the defining representation, the standard coordinate vectors $e_1,e_2,e_3,e_4,e_5$ are weight vectors of weights ${\epsilon }_1,{\epsilon }_2,-{\epsilon }_1,-{\epsilon }_2,0$, respectively. Thus the weight diagram has four axial points and the origin, each of multiplicity one.

For the Adjoint representation, the eight nonzero weights of a representation are the roots from part (ii), each of multiplicity one, with the displayed $X_{\alpha }$ as basis. The zero weight space is the Cartan subalgebra, with basisThe Adjoint representation diagram therefore has the four axial points, four diagonal points, and the origin of multiplicity two. The multiplicities sum to ten, as they must.

For a finite-dimensional Lie algebra, the Killing form is the symmetric bilinear formIn the root-space decomposition of this semisimple Lie algebra, $\mathrm{ad}H$ acts as zero on the abelian Cartan subalgebra and as multiplication by $\alpha (H)$ on ${\mathfrak{g}}_{\alpha }$. For $H,K\in \mathfrak{h}$, their product thus acts as multiplication by $\alpha (H)\alpha (K)$ on each root space. Every root space here is one-dimensional. Taking the trace over the two-dimensional Cartan subalgebra and the eight root spaces provesFor a general weight decomposition the trace would include the corresponding weight multiplicities; the root multiplicities are one in the semisimple setting here.

The PDF's specified five-by-five diagonal matrix is $H=H_1$, so ${\epsilon }_1(H)=1$ and ${\epsilon }_2(H)=0$. In the root sum for the Killing form, the two short roots $\pm {\epsilon }_1$ contribute $1+1$, the two short roots $\pm {\epsilon }_2$ contribute zero, and the four long roots $\pm {\epsilon }_1\pm {\epsilon }_2$ each contribute one. ThereforeThis computation uses the original PDF matrix; the converted TeX incorrectly gives its dimensions.