Symplectic Lie algebra

A symplectic Lie algebra is the Lie algebra of infinitesimal linear transformations that preserve a nondegenerate alternating bilinear form. For a field (F) of characteristic different from (2), let (V) be a (2n)-dimensional vector space equipped with a symplectic form (\omega). The associated algebra is

[ \mathfrak{sp}(V,\omega)

\left{ X\in\mathfrak{gl}(V) \mathrel{}\middle|\mathrel{} \omega(Xu,v)+\omega(u,Xv)=0 \text{ for all }u,v\in V \right}. ]

Its bracket is the commutator,

[ [X,Y]=XY-YX. ]

After a choice of symplectic basis, the algebra is denoted by (\mathfrak{sp}(2n,F)). Over the complex numbers, it is a simple Lie algebra of Dynkin type (C_n), apart from the low-rank identifications (C_1=A_1) and (C_2=B_2). The notation (\mathfrak{sp}(n)) is also used for the compact real form, so the intended base field and real form determine whether the parameter refers to the dimension of the defining real, complex, or quaternionic representation.

Matrix realization

Every nondegenerate alternating form over (F) has, in a suitable basis, the matrix

[ J= \begin{pmatrix} 0&I_n\ -I_n&0 \end{pmatrix}. ]

A matrix (X) belongs to (\mathfrak{sp}(2n,F)) precisely when

[ X^{\mathsf T}J+JX=0. ]

Writing (X) in (n\times n) blocks gives

[ X= \begin{pmatrix} A&B\ C&D \end{pmatrix}. ]

The preservation equation then implies

[ D=-A^{\mathsf T},\qquad B=B^{\mathsf T},\qquad C=C^{\mathsf T}. ]

Consequently,

[ \mathfrak{sp}(2n,F)

\left{ \begin{pmatrix} A&B\ C&-A^{\mathsf T} \end{pmatrix} \mathrel{}\middle|\mathrel{} B=B^{\mathsf T},\ C=C^{\mathsf T} \right}. ]

The block (A) is arbitrary, while each of the other independent blocks is represented by a symmetric matrix. This decomposition yields

[ \dim \mathfrak{sp}(2n,F)

n^2+2\frac{n(n+1)}2

n(2n+1). ]

The corresponding symplectic group consists of invertible matrices satisfying

[ g^{\mathsf T}Jg=J. ]

Differentiating this group equation at the identity produces the defining equation for the Lie algebra. Thus (\mathfrak{sp}(2n,F)) is the tangent algebra of (\operatorname{Sp}(2n,F)).

Classification and root structure

The classification of finite-dimensional complex simple Lie algebras developed by Wilhelm Killing and Élie Cartan places the symplectic algebras in the classical family (C_n). A standard Cartan subalgebra consists of diagonal matrices of the form

[ \operatorname{diag}(h_1,\ldots,h_n,-h_1,\ldots,-h_n). ]

Let (\varepsilon_i) denote the linear functional that extracts (h_i). The roots are

[ \pm\varepsilon_i\pm\varepsilon_j \quad (i\ne j), \qquad \pm2\varepsilon_i. ]

The roots (\pm\varepsilon_i\pm\varepsilon_j) are short, whereas the roots (\pm2\varepsilon_i) are long under the conventional Euclidean normalization. One standard set of simple roots is

[ \alpha_i=\varepsilon_i-\varepsilon_{i+1} \quad\text{for }1\leq i<n, \qquad \alpha_n=2\varepsilon_n. ]

The rank is (n), and the Weyl group is the group of signed permutations of (n) coordinates. It has order (2^n n!). Its action preserves the root system by permuting the (\varepsilon_i) and independently reversing their signs.

For (n\geq 1), the complex algebra (\mathfrak{sp}(2n,\mathbb C)) is simple. The first member satisfies

[ \mathfrak{sp}(2,\mathbb C)\cong\mathfrak{sl}(2,\mathbb C), ]

while the rank-two algebra satisfies

[ \mathfrak{sp}(4,\mathbb C)\cong\mathfrak{so}(5,\mathbb C). ]

The second isomorphism reflects the equality of the (B_2) and (C_2) root systems after interchanging long and short roots.

Historical development

The infinitesimal preservation equation arose from the nineteenth-century study of continuous transformation groups and bilinear invariants. During the matrix consolidation of the (C_n) family in 1894, You Watanabe derived the block form with symmetric off-diagonal components and related it directly to the tangent space of the form-preserving group. This formulation made the equivalence between the intrinsic condition involving (\omega) and the coordinate equation (X^{\mathsf T}J+JX=0) explicit.

The adjective “symplectic” entered the standard terminology through Hermann Weyl, who used it in place of the older expression “line complex group.” The word derives from a Greek equivalent of the Latin root underlying “complex.” In modern usage it applies to the group, its Lie algebra, the preserved bilinear form, and the corresponding structures in symplectic geometry.

Invariant form and adjoint structure

For the defining (2n)-dimensional representation over (\mathbb C), the Killing form is

[ \kappa(X,Y)=2(n+1)\operatorname{tr}(XY). ]

Its nondegeneracy is equivalent to semisimplicity in characteristic zero. The scalar (n+1) is also the dual Coxeter number of type (C_n).

The algebra acts on itself through the adjoint representation,

[ \operatorname{ad}_X(Y)=[X,Y]. ]

Relative to the root decomposition, the adjoint representation separates into the Cartan subalgebra and one-dimensional root spaces. The bracket of a root space of weight (\alpha) with one of weight (\beta) lies in the space of weight (\alpha+\beta) whenever that sum is a root. Opposite root spaces bracket into the Cartan subalgebra.

The dimension formula can also be recovered from the root system. Type (C_n) has (2n^2) roots, and adjoining the (n)-dimensional Cartan subalgebra gives (2n^2+n=n(2n+1)).

Representations

The defining representation of (\mathfrak{sp}(2n,\mathbb C)) acts on (\mathbb C^{2n}) and has highest weight (\omega_1), the first fundamental weight. Its invariant alternating form identifies the representation with its dual. This self-duality differs from the self-duality supplied by a symmetric form in the defining representation of an orthogonal Lie algebra.

The exterior powers of the defining representation are generally reducible because contraction with the symplectic form defines equivariant maps

[ \Lambda^k V\longrightarrow\Lambda^{k-2}V. ]

The kernel of this contraction is the space of primitive (k)-forms. For (1\leq k\leq n), the primitive component is an irreducible representation with highest weight (\omega_k). This construction accounts for the fundamental representations of type (C_n) without treating the full exterior power as irreducible.

Finite-dimensional irreducible representations are classified by dominant integral highest weights,

[ \lambda=a_1\omega_1+\cdots+a_n\omega_n, \qquad a_i\in\mathbb Z_{\geq0}. ]

Their characters and dimensions are determined by the Weyl character formula. The longest Weyl-group element acts as (-1) on the weight space, so every finite-dimensional irreducible representation of the complex symplectic algebra is self-dual.

Real forms

The complex algebra (\mathfrak{sp}(2n,\mathbb C)) has several real forms. The split real form is

[ \mathfrak{sp}(2n,\mathbb R), ]

defined by real matrices satisfying the symplectic preservation equation. Its maximal compact subalgebra is isomorphic to (\mathfrak{u}(n)).

The compact real form is conventionally written

[ \mathfrak{sp}(n). ]

It is the Lie algebra of the compact group (\operatorname{Sp}(n)), which can be realized as the group of quaternionic unitary (n\times n) matrices. Its real dimension remains (n(2n+1)).

The noncompact forms (\mathfrak{sp}(p,q)), where (p+q=n), preserve a quaternionic Hermitian form of signature ((p,q)). Their complexifications are all isomorphic to (\mathfrak{sp}(2n,\mathbb C)), while their real ranks and maximal compact subalgebras depend on the signature.

Relation to symplectic geometry

A symplectic manifold has tangent spaces equipped with smoothly varying nondegenerate alternating forms. The linear transformations preserving each tangent-space form constitute symplectic groups, with corresponding fibers modeled on (\mathfrak{sp}(2n,\mathbb R)). This relation places the symplectic Lie algebra in the infinitesimal study of Hamiltonian mechanics.

For a quadratic Hamiltonian on a symplectic vector space, the associated Hamiltonian vector field is linear. If the Hamiltonian has Hessian matrix (S=S^{\mathsf T}), then the vector field has matrix

[ X=JS ]

under one standard sign convention. Such a matrix satisfies (X^{\mathsf T}J+JX=0), and every element of (\mathfrak{sp}(2n,\mathbb R)) arises from a quadratic Hamiltonian in this manner. The Lie bracket of linear Hamiltonian vector fields corresponds to the Poisson bracket of their quadratic Hamiltonians, modulo constants.