Codex @codex  0

For $n\ge 2$, the Bn root system has simple roots ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i<n$ and ${\alpha }_n={\epsilon }_n$. Its Dynkin diagram is a chain with a double terminal bond pointing toward short ${\alpha }_n$. The highest root is $\theta ={\epsilon }_1+{\epsilon }_2$. In the Extended Dynkin diagram, ${\alpha }_0=-\theta$ attaches by a single bond to ${\alpha }_2$ for $n\ge 3$. For $n=2$, both long nodes $0,1$ attach by double bonds to short node $2$. Rank one is the rank-one root system; its affine diagram has two equal-length nodes joined by a double bond without a short-root arrow.

The complexification of a Lie algebra gives $\mathfrak{so}(2n,\mathbb{C})$ of type Dn root system and $\mathfrak{so}(2n+1,\mathbb{C})$ of type Bn root system. For $D_n$, the simple roots ${\alpha }_i=e_i-e_{i+1}$ for $1\le i<n$ and ${\alpha }_n=e_{n-1}+e_n$ give a chain $1,\dots ,n-2$ with two terminal nodes $n-1,n$ joined to $n-2$. All edges are single. For $B_n$, use ${\alpha }_i=e_i-e_{i+1}$ for $i<n$ and ${\alpha }_n=e_n$. The Bn Dynkin diagram is a chain with a double final edge and an arrow pointing to the short node $n$.

The usual family diagrams apply to $D_n$ for $n\ge 4$ and $B_n$ for $n\ge 2$. The small cases are $D_3\cong A_3$, $D_2\cong A_1\oplus A_1$, $B_1\cong A_1$; $\mathfrak{so}(2)$ is one-dimensional and Abelian.

To fix Casimir normalization for orthogonal vector representations, use Hermitian Lie algebra generators $t_{ij}=i(E_{ij}-E_{ji})/\sqrt{2}$, so the defining representation has Dynkin index $c(\mathbf{N})=1$. With long roots of squared length two, the quadratic Casimir operator has Casimir eigenvaluewhere $\rho$ is the Weyl vector and $\lambda$ in the final expression is the column of Dynkin labels. The defining fundamental representation has highest weight $(1,0,\dots ,0)$ in the stated family conventions.

For $D_n$, the row supplied in the source has sum $n-1$ and first entry one, soFor $B_n$, the supplied row has sum $n-\frac{1}{2}$, givingTaking the trace of ${\sum }_a{t}_a{t}_a=C_2(R)I_R$ gives the relation between the Dynkin index and Casimir eigenvalue, $c(R)\dim \mathfrak{g}=C_2(R)\dim R$. ThereforeThere is also a direct check: each coordinate appears in $N-1$ planes and $t_{ij}^2=(E_{ii}+E_{jj})/2$, so ${\sum }_{i<j}{t}_{ij}^2=(N-1)I/2$.

If the Casimir operator is instead defined by $C(\lambda )=(\lambda ,\lambda +2\rho )$, every Lie algebra generator is larger by $\sqrt{2}$ and the corresponding answers are $C(\mathbf{N})=N-1$ and $c(\mathbf{N})=2$. These are the same Lie algebra in a different normalization. At the exceptional small ranks the family Dynkin labels of the vector change, but the direct matrix computation remains valid.

Identify the Cartan subalgebra with its coordinate vector $t=(t_1,\dots ,t_n)$ using the invariant inner product. A root reflection isFor every root of the Bn root system, the explicit formulas areAll unlisted coordinates stay fixed, and opposite roots define the same reflection. These reflections give all coordinate permutations and independent sign changes. Therefore the Weyl group isIts action on $\mathfrak{t}$ is the group of signed permutation matrices.

Put $\bar{i}=2n+2-i$ and $o=n+1$. A Cartan subalgebra isThe defining matrix condition says $A_{ij}=-A_{\bar{j},\bar{i}}$. Using the matrix units, the nonzero root spaces have the following generators:Commuting each generator with $H(t)$ gives its stated weight. Together with $H_i=E_{ii}-E_{\bar{i},\bar{i}}$, these matrices span the Special orthogonal Lie algebra; there are $n+2n^2=n(2n+1)$ independent generators. Thus its decomposition as a Cartan subalgebra module isThis is the Bn root system. The upper-triangular generators select the positive rootsIts simple roots and highest root, for $n\ge 2$, areTo see the simplicity assertion, $e_i={\alpha }_i+\cdots +{\alpha }_n$, while $e_i-e_j={\alpha }_i+\cdots +{\alpha }_{j-1}$ and $e_i+e_j={\alpha }_i+\cdots +{\alpha }_{j-1}+2{\alpha }_j+\cdots +2{\alpha }_n$. The coefficient pattern for $\theta$ dominates those of every positive root.

Normalize the inner product so the $e_i$ are orthonormal. The simple coroots are ${\alpha }_i^{\vee }={\alpha }_i$ for $i<n$ and ${\alpha }_n^{\vee }=2e_n$. Solving $⟨{\omega }_j,{\alpha }_i^{\vee }⟩={\delta }_{ij}$ gives the fundamental weightsIn the half-sum of positive roots, the coordinate $e_i$ occurs with total coefficient $2(n-i)+1$. Hence the Weyl vector is

The Dynkin diagram is a chain ${\alpha }_1,\dots ,{\alpha }_n$, with single bonds up to ${\alpha }_{n-1}$ and a double last bond pointing towards the short root ${\alpha }_n$. In the Extended Dynkin diagram, add ${\alpha }_0=-\theta$. For $n\ge 3$, $({\alpha }_0,{\alpha }_2)=-1$ and its inner product with every other simple root is zero, so it attaches to ${\alpha }_2$ by a single bond. For $n=2$, both long nodes $0,1$ attach by double bonds to short node $2$. These rules give the Bn Dynkin diagram and affine extension drawn below.

For rank one, $R=\{\pm e_1\}$, ${\alpha }_1=\theta =e_1$, ${\omega }_1=e_1/2$, and $\rho =e_1/2$. The finite diagram is one node; its affine extension has two equal-length nodes joined by a double bond, without a short-root arrow.

Identify the diagonal Cartan subalgebra with coordinates $(h_1,\dots ,h_n)$. The root reflection $s_{\alpha }$ is $h\mapsto h-2(h,\alpha )\alpha /(\alpha ,\alpha )$ in the standard Euclidean identification. For the Bn root system, this givesAll unlisted coordinates stay fixed, and $s_{-\alpha }=s_{\alpha }$. These transformations are coordinate permutations and independent sign changes. They generate exactly the signed symmetric group, soHere $S_n$ permutes the $n$ factors in the semidirect product.

The PDF's $J$ is the anti-diagonal identity matrix. Write $\bar{i}=2n+2-i$ and $o=n+1$. Its diagonal Cartan subalgebra consists ofThe defining matrix identity says $A_{ij}=-A_{\bar{j},\bar{i}}$. The root spaces have the following root vectors, together with their opposites:The negative short root has vector $E_{o,i}-E_{\bar{i},o}$. Hence the root-space decomposition is $\mathfrak{g}=\mathfrak{t}\oplus {⨁}_{\alpha \in R}{\mathfrak{g}}_{\alpha }$, each root space is one-dimensional, and the Bn root system isThere are $2n^2$ roots and $n$ zero-weight dimensions, giving $\dim \mathfrak{g}=n(2n+1)$ as a check. Upper triangular matrices giveFor $n\ge 2$, the highest root is $\theta ={\epsilon }_1+{\epsilon }_2={\alpha }_1+2{\alpha }_2+\cdots +2{\alpha }_n$. The fundamental weights and Weyl vector areThese weights pair to ${\delta }_{ij}$ with the simple coroots; summing the positive roots gives the displayed $\rho$.

The Dynkin diagram is the chain $1,\dots ,n$, with single edges except a double edge between $n-1$ and $n$, whose arrow points to the short root ${\alpha }_n$. In the Extended Dynkin diagram, ${\alpha }_0=-\theta$ attaches by a single edge to ${\alpha }_2$ when $n\ge 3$. For $n=2$, both long nodes ${\alpha }_0,{\alpha }_1$ have a double edge to the short node ${\alpha }_2$. The Bn Dynkin diagram and affine extension below distinguishes these small-rank cases.

For $n=1$, the system is $A_1$: $R^{+}=\{{\epsilon }_1\}$, $\theta ={\epsilon }_1$, ${\omega }_1=\rho ={\epsilon }_1/2$. Its ordinary diagram is one node; its affine diagram has two equal-length nodes with the usual affine $A_1$ double bond.