Symplectic Nilpotent Lie Groups

• Symplectic nilpotent Lie groups are connected, simply connected nilpotent Lie groups equipped with nondegenerate, closed 2-forms that serve as key examples in differential and symplectic geometry.
• Their structure is characterized by intricate lattice classifications, cohomological obstructions, and reduction processes, leading to diverse non-Kähler compact symplectic nilmanifolds.
• Applications include constructing explicit affine and symplectic nilmanifolds, employing Lagrangian extension theory, and advancing geometric quantization through double Lie group frameworks.

A symplectic nilpotent Lie group is a connected, simply connected nilpotent Lie group equipped with a left invariant symplectic form—i.e., a nondegenerate, closed, left invariant 2-form. These objects constitute a fundamental class of examples in differential, algebraic, and geometric representation theory, geometric group theory, and the construction of compact non-Kähler symplectic manifolds. Their structure, classification, and invariants are governed by deep interactions between Lie algebra cohomology, affine and symplectic geometry, isotropic subgroup structure, and group-theoretic discretization via lattices.

1. Lattices, Affine and Symplectic Structures

A central differentiating feature of symplectic nilpotent Lie groups is the abundance and classification of lattices (i.e., discrete, co-compact subgroups) compatible with the symplectic structure (Medina et al., 2012). In many cases, notably for 2-step nilpotent groups such as various realizations of the Heisenberg group $H_1(\mathbb F)$ over $\mathbb F = \mathbb R,\,\mathbb C,\,\mathbb D$, there exist infinite families of pairwise noncommensurable lattices, yielding infinitely many nonhomeomorphic compact nilmanifolds.

A typical 6-dimensional nilpotent Lie algebra arising in this context is presented on a basis $\{e_1,\ldots,e_6\}$ as

$$[e_1, e_2] = [e_3, e_4] = e_5, \quad [e_1, e_3]=e_6, \quad [e_2,e_4] = -d e_6,$$

with $d$ a squarefree integer; the distinct values of $d$ distinguish inequivalent rational structures and lattice types. The cohomological structure of such Lie algebras is analyzed using left-invariant symplectic forms and flat, torsion-free affine connections, which are constructed via relations of the form

$\omega(a\cdot b, c) = -\omega(b, [a, c])$

for a scalar 2-cocycle $\omega$ and left-symmetric algebra product "$\cdot$". These invariants transmit to the compact quotients, producing (infinite families of) compact affine and symplectic nilmanifolds.

2. Cohomological and Obstruction Theory

Necessary and sufficient conditions for the existence of left-invariant symplectic forms on (nilpotent) Lie algebras are governed by fine cohomological obstructions (Barco, 2012). The key invariant is the $(0,2)$ intermediate cohomology group $E^\infty_{(0,2)}(\mathfrak g)$ arising from a canonical filtration of the Chevalley–Eilenberg complex. For a nilpotent Lie algebra $\mathfrak g$, if $E^\infty_{(0,2)}(\mathfrak g)=0$, then $\mathfrak g$ and any trivial extension $\mathfrak g\oplus \mathbb R^s$ do not admit symplectic structures. This obstruction rules out symplectic forms on essentially all nilradicals of minimal parabolic subalgebras of split real forms of classical simple Lie algebras, with only low-dimensional exceptions (corresponding to abelian, 3D Heisenberg, and $so(5,\mathbb R)$-nilradicals).

This analysis not only provides negative existence results but, in positive cases, relates the $(0,2)$ component to explicit, nondegenerate closed 2-forms whose cohomology class “jumps” out of the lower pieces of the filtration, ensuring nontriviality.

3. Structure Theory: Symplectic Reduction and Lagrangian Extensions

Symplectic nilpotent Lie groups admit a rich reduction theory via sequences of reductions by isotropic normal subgroups, yielding a unique irreducible symplectic base at the terminus (Baues et al., 2013). Under this lens, cotangent symplectic Lie groups and Lagrangian extension theory provide an organizing principle: every simply connected flat (i.e., affine) Lie group can be realized as a Lagrangian subgroup of a symplectic nilpotent Lie group of cotangent type, and conversely, symplectic nilpotent Lie groups with Lagrangian ideals are Lagrangian extensions (“cotangent extensions”) of lower-dimensional flat, nilpotent Lie algebras.

Given a flat, geodesically complete nilpotent Lie algebra $\mathfrak h$ with flat connection $\nabla$, one constructs the symplectic nilpotent Lie algebra

$$\mathfrak g = \mathfrak h \oplus_\rho \mathfrak h^*$$

with Lie bracket

$$[(x,\xi), (y,\eta)] = ([x,y]_\mathfrak h, \rho(x)\eta - \rho(y)\xi + \alpha(x,y)),$$

for a 2-cocycle $\alpha$ (satisfying a cyclic Bianchi-type identity) and dual representation $\rho$. The canonical symplectic form is given by the dual pairing. These data are classified up to isomorphism by the “Lagrangian extension cohomology” group $H^2_{L,\rho}(\mathfrak h, \mathfrak h^*)$.

Explicitly, an eight-dimensional symplectic nilpotent Lie group with Lagrangian normal subgroup $\mathcal J$ is classified by a quadruple $(\mathcal H, \nabla, [\alpha])$, where $\mathcal H$ is a 4-dimensional flat nilpotent Lie group, $\nabla$ a flat, complete connection, and $[\alpha]\in H^2_{L,\rho}(\mathcal H, \mathcal H^*)$. The classification in (Aissa et al., 23 Aug 2025) identifies exactly 95 such non-isomorphic cases.

4. Heisenberg Groups over Algebras and Symplectic Forms

For Heisenberg-type nilpotent Lie groups, a complete characterization of the existence of left-invariant symplectic structures over local associative and commutative finite-dimensional real algebras $A$ is available (Medina et al., 2012). The Lie algebra $H_1(A)$ admits a nondegenerate scalar 2-cocycle if and only if $\dim_\mathbb R A$ is even and the dimension $d$ of the socle (annihilator of the maximal ideal) satisfies $d\leq 2$. In the case $d=1$, the algebra is Frobenius, and explicit symplectic forms can be constructed. For $d=2$, a more delicate balancing of cocycle terms is required. General $k$-step Heisenberg algebras $H_k(A)$ ($k\geq 2$) do not admit nondegenerate cocycles, so only the "minimal" cases yield symplectic nilpotent Lie groups.

Furthermore, lattices in such Heisenberg groups lift canonically to lattices in the corresponding double Lie group associated to solutions of the classical Yang-Baxter equation, and the structure extends to the context of Manin triples and double Lie algebra constructions.

5. Compact Symplectic Nilmanifolds and Geometric Quantization

Given a lattice $\Gamma$ in a symplectic nilpotent Lie group $G$, the quotient $M = \Gamma \backslash G$ forms a compact nilmanifold carrying a left-invariant symplectic structure. These manifolds frequently admit complete flat affine connections compatible with the symplectic structure, and their induced Poisson structures are often polynomial. Central (Hamiltonian) group actions allow the application of symplectic reduction—producing lower-dimensional affine-symplectic nilmanifolds, whose moment images can be polytopal (Delzant-type), and which inherit both affine and symplectic structures naturally from the reduction procedure.

Such structures play an important role in geometric quantization, as the existence of Lagrangian (normal) subgroups supplies the required polarizations. The cotangent construction relates the geometry of nilpotent Lie groups to the representation theory of their affine and symplectic automorphism groups.

6. Mathematical Formulations and Explicit Structure

Key algebraic structures and relations defining symplectic nilpotent Lie groups include:

• The canonical left-symmetric product and associated flat torsion-free connection:

$\omega(a \cdot b, c) = -\omega(b,[a,c]),$

with $[a,b]=a\cdot b - b\cdot a$.

• In the Lagrangian extension model for $$\mathfrak g = \mathfrak h \oplus_\rho \mathfrak h^*$$, the Lie bracket

$$[(x,\xi),(y,\eta)] = ([x,y]_\mathfrak h,\, \rho(x)\eta-\rho(y)\xi + \alpha(x,y))$$

and the dual action $\rho(x)\xi = -\xi\circ \nabla_x$.

• Cocycles $\alpha$ satisfy the compatibility:

$$\alpha(x,y)(z) + \alpha(y,z)(x) + \alpha(z,x)(y) = 0,\quad \forall x,y,z.$$

• The condition for Heisenberg Lie algebra $H_1(A)$ to admit a symplectic form: $A$ even-dimensional, $\dim S \leq 2$; symplectic form explicitly,

$$\omega(e\otimes a + f\otimes b + g\otimes c,\, e\otimes a' + f\otimes b' + g\otimes c') = \pi(ac' - a'c) + \omega'(b,b')$$

for $\pi$, a linear form on $A$, and $\omega'$ an alternating form.

7. Consequences and Open Directions

The explicit algebraic and cohomological models for symplectic nilpotent Lie groups have several notable consequences:

• The classification of compact nilmanifolds with left-invariant symplectic (and often affine) structures—providing a rich class of non-Kähler compact symplectic manifolds.
• The specification and count of Lagrangian normal subgroups, especially in low dimensions: for eight-dimensional symplectic nilpotent Lie groups, exactly 95 distinct isomorphism classes with Lagrangian normal subgroups exist (Aissa et al., 23 Aug 2025).
• Construction of infinite families of non-homeomorphic compact affine/symplectic nilmanifolds distinguished by rational invariant data (e.g., squarefree parameters in nilpotent Lie algebra brackets).
• Frameworks for constructing double Lie groups and Manin triples arising from symplectic nilpotent settings, with direct connections to solutions of the classical Yang–Baxter equation.

Further research and open questions include extending these classification techniques to higher dimensions, describing symplectic nilpotent Lie groups without Lagrangian subgroups, and analyzing the impact of additional geometric structures (complex, Kähler, contact) built over the symplectic nilpotent setting.