Symplectic Lie algebra (${\mathfrak{sp}}_{2n}$)

For a nondegenerate alternating matrix $J$, the symplectic Lie algebra is ${\mathfrak{sp}}_{2n}=\{A:AJ+J{A}^T=0\}$.

Relative to $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$, a symplectic Lie algebra matrix has blocks $\left(\begin{array}{cc}P& Q\\ R& -P^{\mathsf{T}}\end{array}\right)$ with $Q,R$ symmetric. Its diagonal Cartan subalgebra has basis $E_{ii}-E_{l+i,l+i}$. Write ${\epsilon }_i(h)={\lambda }_i$. A root-space decomposition is obtained fromThe matrix unit rule $[h,E_{ab}]=(h_{aa}-h_{bb})E_{ab}$ proves $[h,e_r]=r(h)e_r$. These root vectors and the diagonal basis span all the permitted blocks and are independent. There are $2l^2$ roots and $l$ diagonal basis elements, giving dimension $l(2l+1)$.

For $n\ge 2$, the defining Lie algebra representation $V$ of ${\mathfrak{sp}}_{2n}$ has the displayed tensor product decomposition. The crystal of the defining symplectic representation and the crystal tensor-product rule give exactly these three highest vertices, each occurring once. The symmetric square is $L(2{\omega }_1)$ of dimension $n(2n+1)$. The exterior square splits into its invariant inverse-form line and the primitive exterior square $L({\omega }_2)$ of dimension $n(2n-1)-1$. Their dimensions sum to $4n^2$. At $n=1$, the exterior square is only the invariant line, giving $L(2{\omega }_1)\oplus L(0)$.

The Lie algebra of the compact symplectic group consists of $X^{†}=-X$ and $XJ+J{X}^T=0$. Equivalently $X=\left(\begin{array}{cc}A& B\\ -\overline{B}& \overline{A}\end{array}\right)$ with $A^{†}=-A$ and $B^T=B$. Its real dimension is $n(2n+1)$, and its complexification is the symplectic Lie algebra.

In its block form, a matrix-unit basis of the unitary Lie algebra spans the anti-Hermitian block $A$, with the corresponding conjugate block below. Independent real and imaginary symmetric matrix units span $B$, with lower block $-\overline{B}$. These give $n^2+n(n+1)$ independent real generators, including diagonal symmetric entries.

For the symplectic Lie algebra $\mathfrak{sp}(4)$ preserving $J=\left(\begin{array}{cc}0& I_2\\ -I_2& 0\end{array}\right)$, write $h_1=E_{11}-E_{33}$ and $h_2=E_{22}-E_{44}$. Its positive roots of a root system have sl2 subalgebra associated with a root triplesThe matrix unit commutator identity verifies $[X_{\alpha },Y_{\alpha }]=H_{\alpha }$ and the two eigenvalue relations $[H_{\alpha },X_{\alpha }]=2X_{\alpha }$, $[H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }$.

Over $\mathbb{C}$, the symplectic Lie algebra on a four-dimensional symplectic vector space acts faithfully on its five-dimensional primitive exterior square, preserving its symmetric bilinear form. Faithfulness follows because an operator killing all wedges $e_i\wedge e_j$ has all off-diagonal coefficients zero and satisfies $A_{ii}+A_{jj}=0$ for every pair, hence is zero. The source and target Lie algebras both have dimension ten, so this faithful map is an isomorphism to the Special orthogonal Lie algebra ${\mathfrak{so}}_5$.

For the defining four-dimensional representation $V$ of the symplectic Lie algebra ${\mathfrak{sp}}_4$, with the short simple root numbered first, its highest weight is ${\omega }_1$. Its tensor square decomposes aswith dimensions $10,5,1$. The first summand is the symmetric square; the other two form the exterior square, split by symplectic contraction of an exterior square.

In an orthonormal basis $({\epsilon }_i)$, the $C_n$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $i<j$ and $\pm 2{\epsilon }_i$.

For $n\ge 2$, the finite Dynkin diagram has a chain of short roots ${\alpha }_1,\dots ,{\alpha }_{n-1}$ with single bonds, followed by a double bond to the long root ${\alpha }_n$, pointing toward ${\alpha }_{n-1}$. The Extended Dynkin diagram adds the long root ${\alpha }_0=-2{\epsilon }_1$, doubly bonded to ${\alpha }_1$ with arrow toward ${\alpha }_1$. All other bonds remain. For $n=1$, the finite diagram is a single vertex; the affine diagram has two equal-length vertices joined by a double bond without an arrow, with Cartan matrix $\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)$.

Choose positive roots of a root system ${\epsilon }_i-{\epsilon }_j$, ${\epsilon }_i+{\epsilon }_j$ for $i<j$, and $2{\epsilon }_i$. The simple roots are ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i<n$ and ${\alpha }_n=2{\epsilon }_n$. Their coroots are ${\alpha }_i^{\vee }={\epsilon }_i-{\epsilon }_{i+1}$ and ${\alpha }_n^{\vee }={\epsilon }_n$. Therefore the fundamental weights and Weyl vector areThe highest root is $2{\epsilon }_1=2{\alpha }_1+\cdots +2{\alpha }_{n-1}+{\alpha }_n$. These formulas include rank one.

The Weyl group of the Cn root system is the signed symmetric group. A Weyl reflection in $2{\epsilon }_i$ changes the sign of coordinate $i$; one in ${\epsilon }_i-{\epsilon }_j$ interchanges coordinates $i,j$; one in ${\epsilon }_i+{\epsilon }_j$ interchanges and negates both coordinates. These generate all independent sign changes and all permutations, giving order $2^{n}n!$.

In an orthonormal basis $({\epsilon }_1,{\epsilon }_2)$, the roots are $\pm ({\epsilon }_1-{\epsilon }_2)$, $\pm ({\epsilon }_1+{\epsilon }_2)$, $\pm 2{\epsilon }_1$, and $\pm 2{\epsilon }_2$. Choose simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$ and ${\alpha }_2=2{\epsilon }_2$, with ${\alpha }_1$ a short root. The fundamental weights are ${\omega }_1={\epsilon }_1$ and ${\omega }_2={\epsilon }_1+{\epsilon }_2$.

The $C_3$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $1\le i<j\le 3$ and $\pm 2{\epsilon }_i$ for $1\le i\le 3$. One root basis is ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2-{\epsilon }_3$, ${\alpha }_3=2{\epsilon }_3$. With $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$, its Cartan matrix isThe fundamental representation of highest weight $[1,0,0]$ is six-dimensional, with weights $\pm {\epsilon }_1,\pm {\epsilon }_2,\pm {\epsilon }_3$.

In the realization $\Phi (C_3)=\{\pm 2e_i,\pm e_i\pm e_j:i<j\}$, each coordinate two-plane cuts out a C2 root system. Each plane $\pm x_1\pm x_2\pm x_3=0$ cuts out an A2 root system; there are four such planes after identifying opposite normals. Thus the irreducible full rank-two root subsystems comprise three of type $C_2$ and four of type $A_2$. For an $A_2$ simple pair $p,q$, equal lengths and angle $120$ degrees give roots $\pm p,\pm q,\pm (p+q)$. For a mixed-length simple pair $s,l$ with $\Vert l{\Vert }^2=2\Vert s{\Vert }^2$ and $s\cdot l=-\Vert s{\Vert }^2$, the B2 root system has roots $\pm s,\pm l,\pm (s+l),\pm (2s+l)$. These lists also verify that the selected pairs are simple, rather than simply independent.

The C3 root system consists of $\pm 2e_i$ and $\pm e_i\pm e_j$ in ${\mathbb{R}}^3$. Its Weyl group is the signed symmetric group on three coordinates, of order $48$. The reflecting planes are $x_i=0$ and $x_i=\pm x_j$. The open Weyl chamber $x_1>x_2>x_3>0$ has closure generated by $(1,0,0)$, $(1,1,0)$ and $(1,1,1)$. Sign choices and absolute-coordinate orderings give its $48$ translates. On the unit sphere the chamber is a spherical triangle with angles $\pi /2$, $\pi /3$, $\pi /4$, and hence area $\pi /12$; the areas of all $48$ triangles sum to $4\pi$.

For the C3 root system, take ${\alpha }_1=e_1-e_2$, ${\alpha }_2=e_2-e_3$, ${\alpha }_3=2e_3$. With columns indexing coroots, the Cartan matrix isThe convention with rows indexing coroots gives $A^T$. Both describe a double bond between the short root ${\alpha }_2$ and long root ${\alpha }_3$, with arrow toward ${\alpha }_2$.