Tangent-Space Differentiation on Lie Groups

• Tangent-space differentiation on Lie groups is a framework that integrates differential, symplectic, and algebraic tools to handle derivatives on non-linear, non-commutative manifolds.
• The approach utilizes flat affine connections and deformed (co)tangent bundles through left-symmetric and post-left-symmetric algebras to build hypersymplectic and special symplectic structures.
• These constructions facilitate explicit formulations of geometric deformations and algebraic extensions with significant applications in integrable systems, quantum geometry, and mathematical physics.

Tangent-space differentiation on Lie groups encompasses the set of geometric, algebraic, and analytic tools used to formulate, manipulate, and compute derivatives (or differential structures) that are compatible with the non-linear, non-commutative structure of Lie groups. In the context of "Special symplectic Lie groups and hypersymplectic Lie groups" (Ni et al., 2010), tangent-space differentiation is deeply intertwined with symplectic and affine geometry, the theory of left-symmetric and post-left-symmetric algebras, and the construction of hypersymplectic structures on (co)tangent bundles. This synthesis describes the foundational concepts, algebraic frameworks, geometric deformations, key mathematical formulations, and broad implications of tangent-space differentiation on Lie groups as articulated in this context.

1. Symplectic and Hypersymplectic Structures

A symplectic Lie group is a finite-dimensional real Lie group $G$ together with a left-invariant, non-degenerate 2-form $\omega$ that is closed. A special symplectic Lie group is a triple $(G, \omega, \nabla)$, where $G$ is a Lie group, $\omega$ a left-invariant symplectic form, and $\nabla$ a left-invariant, flat, torsion-free affine connection such that $\omega$ is parallel with respect to $\nabla$. The compatibility condition is

$$\omega(\nabla_x y, z) = \omega(\nabla_x z, y), \qquad \forall x, y, z \in \mathfrak{g}$$

with $\mathfrak{g}$ the Lie algebra of $G$.

Hypersymplectic structures generalize hyperkähler structures to neutral signature: a hypersymplectic Lie group possesses a left-invariant complex product structure $\{J, E\}$ and a neutral metric $g$ of signature $(2n,2n)$, such that the forms

$$\omega_1(X,Y) = g(JX,Y),\qquad \omega_2(X,Y) = g(EX,Y),\qquad \omega_3(X,Y) = g(JE X, Y)$$

are each closed and non-degenerate. These compatible structures allow the tangent and cotangent bundles of $G$ to inherit rich geometric properties.

2. Affine Structures and Deformation of Tangent Bundles

A flat, torsion-free left-invariant affine connection $\nabla$ on $G$ yields a compatible left-symmetric algebra (LSA) structure on $\mathfrak{g}$. Let $V(x,y) = \nabla_x y$, so $$V : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$$ induces an LSA if $V$ satisfies:

$$V(x,y) - V(y,x) = [x, y], \ V(x, V(y,z)) - V(y, V(x,z)) = V([x,y],z)$$

for all $x, y, z \in \mathfrak{g}$.

The deformed (co)tangent bundle group structure is given (for the tangent bundle $TG \cong G \times \mathfrak{g}$) by:

$$(g_1, x_1) \cdot (g_2, x_2) = (g_1 g_2, x_1 + f_V(g_1)x_2)$$

where $f_V: G \to \operatorname{Aut}(\mathfrak{g})$ is the homomorphism induced by $V$. The bracket on the corresponding Lie algebra $\mathfrak{g} \ltimes \mathfrak{g}$ is:

$$[(x_1, y_1), (x_2, y_2)] = ([x_1, x_2], V_{x_1}y_2 - V_{x_2}y_1)$$

An analogous deformation occurs for $T^*G$ using the affine structure, yielding a Lie group structure whose geometry encodes the symplectic data.

3. Post-Left-Symmetric Algebras and Cotangent Extensions

The extension problem for cotangent bundles—constructing LSA structures on $\mathfrak{g} \oplus \mathfrak{g}^*$ compatible with a canonical pairing—motivates the introduction of post-left-symmetric algebras (PLSA). A PLSA on a vector space $A$ consists of bilinear operations $<$ (commutative) and $>$ (LSA) such that:

• $(A, >)$ is an LSA,
• $x < y = y < x$ for all $x, y$,
• $x > (y < z) = (x \cdot y) < z + y < (x \cdot z)$ where $x \cdot y := x < y + x > y$.

This splitting of the LSA product into symmetric and non-symmetric parts encapsulates deeper algebraic differentiation and underlies the construction of "double extensions"—Lie groups whose Lie algebras are $\mathfrak{g} \oplus \mathfrak{g}^*$ with simultaneous preservation of symplectic and affine structures. The canonical symplectic form on $\mathfrak{g} \oplus \mathfrak{g}^*$ is:

$$\omega_p((x, a^*), (y, b^*)) = -\langle x, b^* \rangle + \langle a^*, y \rangle$$

4. Central Mathematical Formulas

Key mathematical expressions and constructions used in the paper include:

Context	Formula	Reference
Special symplectic compatibility	$\omega(V(x, y), z) = \omega(V(x, z), y)$	Eq. (4)
Tangent bundle group multiplication	$$(g_1,x_1)\cdot(g_2,x_2) = (g_1g_2, x_1 + f_V(g_1)x_2)$$	Eq. (10)
Semidirect sum Lie algebra bracket	$$[(x_1,y_1),(x_2,y_2)] = ([x_1,x_2], V_{x_1}y_2 - V_{x_2}y_1)$$	Eq. (11)
Natural hypersymplectic metric on $TG$	$g((x, z), (y, w)) = \omega(x,w) + \omega(y,z)$	Eq. (16)
Cotangent extension compatibility (PLSA)	$x > (y < z) = (x \cdot y) < z + y < (x \cdot z)$	Eq. (33)
Canonical pairing on cotangent extension	$$\omega_p((x, a^*), (y, b^*)) = -\langle x, b^*\rangle + \langle a^*, y \rangle$$	Eq. (32)

These structures and relations are essential for constructing Lie groups with hypersymplectic structures and for extending symplectic geometry into the field of affine and Lie-theoretic settings.

5. Applications and Implications

The construction of deformed tangent (and cotangent) bundle Lie groups from special symplectic Lie groups provides a systematic method for generating new examples of hypersymplectic Lie groups, notably increasing the known catalog of such structures. The solution of the affine cotangent extension problem via PLSA yields a conceptual framework unifying affine and symplectic geometry, reminiscent of double and Drinfeld extensions familiar from Poisson–Lie theory and the theory of classical $r$-matrices.

These developments have notable applications:

• In mathematical physics, particularly in integrable systems, string theory, and supersymmetric models, the constructed geometries offer new examples with vanishing Ricci curvature and special holonomy.
• The algebraic framework elucidates the bridge between the linear (tangent) and global (group) structure, making explicit the way tangent-space differentiation is governed by deeper algebraic hierarchies.
• The notion of post-left-symmetric algebras and their double extensions open possible avenues for constructing solutions to the classical Yang–Baxter equation and for studying quantum deformations.

6. Interplay Between Affine, Symplectic, and Algebraic Structures

Tangent-space differentiation, in this context, cannot be separated from the triple mutual compatibility between affine (differentiation), symplectic (non-degenerate 2-forms), and algebraic (post-left-symmetric) structures. The process of deforming the tangent or cotangent bundle via a flat affine structure, while simultaneously controlling the symplectic pairing, is governed by the theory of LSAs and their extensions. The explicit algebraic decomposition of products, and careful compatibility conditions (notably in post-LSA), render tangent-space differentiation a structure-rich process where the geometry of the group is encoded as much in its tangent and cotangent algebraic extensions as in its manifold-level properties.

7. Broader Context and Theoretical Significance

These results fit into a wider landscape in geometric mechanics and Lie theory, where the affine, symplectic, and metric properties of tangent and cotangent bundles of Lie groups play central roles in the paper of reduction, integrable systems, and special geometric structures. The introduction and characterization of post-left-symmetric and compatible LSA extensions as the algebraic backbone of these geometric deformations provide a blueprint for exploring analogous constructions in other settings (e.g., Poisson–Lie groups, double Lie algebras, and quantum geometry).

In conclusion, tangent-space differentiation on Lie groups, as developed through the deformation of (co)tangent bundles of special symplectic Lie groups via compatible affine structures and their associated post-left-symmetric algebras, yields a unified formalism for producing Lie groups with hypersymplectic and other advanced geometric structures. These constructions underpin numerous applications in geometry and physics, and reveal deep algebraic relationships that govern differential, symplectic, and affine structures in the tangent spaces of Lie groups.