Non-Semisimple Lie Algebra gₙ

• Non-semisimple Lie algebra gₙ is defined as a semidirect product of sl₂ with a 2-step nilpotent radical, characterized by a unique determinant-based Casimir invariant.
• The algebra’s representation theory aligns finite-dimensional irreducible modules with sl₂ highest-weight modules, where the nilpotent part acts trivially.
• Its coalgebra and Hopf algebra structures facilitate the construction of commuting integrals, underpinning the classification of integrable Hamiltonian hierarchies.

The non-semisimple Lie algebra $\mathfrak{g}_n$ constitutes a chain of $n(n+1)/2$-dimensional Lie algebras that generalize both $\mathfrak{sl}_2(\mathbb{K})$ and the two-photon Lie algebra $\mathfrak{h}_6$. These structures exhibit a fusion of a semisimple $\mathfrak{sl}_2$ subalgebra and a 2-step nilpotent radical, resulting in distinctive algebraic properties, representation theory, coalgebra symmetries, and integrable Hamiltonian hierarchies. Key results include the existence of a unique nontrivial Casimir element defined as the determinant of an $n\times n$ symmetric matrix, the classification of irreducible representations, explicit coalgebra and Hopf structures, and integrability properties for the associated Hamiltonian systems depending on $n$ (Gubbiotti et al., 1 Dec 2025). For the related case $g_n = \mathfrak{sl}_n \ltimes \mathbb{C}^n$, free symmetric invariant algebras are classified with explicit dependence on rank $n$ (Yakimova, 2015).

1. Algebraic Definition and Structure

$\mathfrak{g}_n$ is defined for $n \geq 2$ with dimension $T_n = n(n+1)/2$. The algebra admits a basis incorporating:

• Three $\mathfrak{sl}_2$ generators: $h$, $x_+$, $x_-$,
• $2(n-2)$ "bosonic" generators: $y_{i,\pm}$ for $i=1,...,n-2$,
• $T_{n-2} = (n-2)(n-1)/2$ central elements: $z_{i,j}$ with $1 \leq i \leq j \leq n-2$.

The nonzero Lie brackets are:

• $[x_+, x_-] = h$, $[h, x_+] = 2x_+$, $[h, x_-] = -2x_-$,
• $[h, y_{i,\pm}] = \pm y_{i,\pm}$,
• $[x_+, y_{i,-}] = y_{i,+}$, $[x_-, y_{i,+}] = y_{i,-}$,
• $[y_{i,+}, y_{j,-}] = z_{\min(i,j), \max(i,j)}$.

The Levi–Malcev decomposition reads:

$$\mathfrak{g}_n = \mathfrak{sl}_2(\mathbb{K}) \ltimes r_n,$$

where the radical $r_n$ is 2-step nilpotent, consisting of $y_{i,\pm}$ and central $z_{i,j}$ elements. The structure is therefore that of a semidirect product between the semisimple $\mathfrak{sl}_2(\mathbb{K})$ part and its nilpotent radical, sometimes termed a generalized Heisenberg algebra (Gubbiotti et al., 1 Dec 2025).

2. Casimir Elements and Invariant Theory

A rank-count via the Beltrametti–Blasi method establishes exactly $T_{n-2}+1$ algebraically independent Casimir elements: $T_{n-2}$ trivial Casimirs corresponding to central $z_{i,j}$ and a unique nontrivial Casimir polynomial of degree $n$. This Casimir is constructed from the determinant of an $n \times n$ matrix $M_n$ whose entries consist of the generators $z_{i,j}$, $y_{i,\pm}$, $x_\pm$, and $h$ as follows:

$$M_n = \begin{pmatrix} z_{1,1} & \cdots & -y_{1,-} & y_{1,+} \ \vdots & \ddots & \vdots & \vdots \ -z_{1,n-2}&\cdots & -y_{n-2,-}& y_{n-2,+} \ -y_{1,-} & \cdots & -2x_- & h \ y_{1,+} & \cdots & h & 2x_+ \end{pmatrix}$$

with the Casimir element given by $C_n = -\det M_n$ (Gubbiotti et al., 1 Dec 2025).

For the related algebra $g_n = \mathfrak{sl}_n \ltimes \mathbb{C}^n$, symmetric invariants consist solely of the $\mathfrak{sl}_n$ Casimirs $\mathrm{Tr}(A^k)$ for $2 \leq k \leq n$ (with $A$ the adjoint) for $n \geq 4$, and an additional "vector-determinant" invariant for $n=3$ (Yakimova, 2015).

3. Representation Theory

Lie's theorem implies that in any finite-dimensional irreducible module for $\mathfrak{g}_n$, the nilpotent radical $r_n$ acts trivially. Consequently,

$$\mathrm{Irr}(\mathfrak{g}_n) \cong \mathrm{Irr}(\mathfrak{sl}_2(\mathbb{K}))$$

via pullback along the projection $\mathfrak{g}_n \to \mathfrak{sl}_2$. Each irreducible representation is thus a highest-weight $\mathfrak{sl}_2$-module $V_m$ (dimension $m+1$), with $y_{i,\pm}$, $z_{i,j}$ acting as zero. Indecomposable modules with nontrivial radical action can be constructed via nilpotent action among Jordan blocks but lack complete classification in current literature (Gubbiotti et al., 1 Dec 2025).

4. Coalgebra Structure and Hopf Algebraic Symmetry

The universal enveloping and symmetric algebras of $\mathfrak{g}_n$ are equipped with standard primitive coalgebra maps:

• $\Delta(x) = x \otimes 1 + 1 \otimes x$ (coproduct),
• $\varepsilon(x) = 0$ (counit),
• $S(x) = -x$ (antipode).

These structures endow $S(\mathfrak{g}_n)$ with Poisson coalgebra symmetry. Iterated coproducts $\Delta_L^{(m)}$, $\Delta_R^{(m)}$ applied to the Casimir $C_n$ yield two families of commuting functions $C_n^{(m)}$ and $C_{n,(m)}$, which remain in involution with themselves and the diagonal algebra. Realizing these via one-particle symplectic representations allows the construction of $N$-particle Hamiltonians and their universal first integrals (Gubbiotti et al., 1 Dec 2025).

5. Hamiltonian Hierarchies and Integrability Classification

Canonical symplectic realizations assign $\mathfrak{g}_n$ generators to polynomials in $(q, p)$ variables and constants:

• $h \mapsto q p$, $x_- \mapsto -q^2/2$, $x_+ \mapsto p^2/2$,
• $y_{i,-} \mapsto -\alpha^{(i)} q$, $y_{i,+} \mapsto \alpha^{(i)} p$,
• $z_{i,j} \mapsto \alpha^{(i)} \alpha^{(j)}$.

The $N$-site Hamiltonian takes the form:

$$H_n^{(N)}(\mathbf{q}, \mathbf{p}) = H_n\left(\sum_i q_i p_i, -\frac{1}{2} \sum_i q_i^2, \frac{1}{2} \sum_i p_i^2, -\sum_i \alpha^{(i)} q_i, \sum_i \alpha^{(i)} p_i, \{\alpha^{(i)} \alpha^{(j)}\} \right)$$

with integrals derived from Casimirs as sums of squares of certain determinants ("building-blocks" $L_{i_1...i_n}$) over $n$-tuples of sites. The integrability type depends on $n$:

• For $n=2$, the system is Liouville-integrable,
• $n=3$: quasi-integrable, Poincaré–Lyapounov–Nekhoroshev type (rank $N-1$),
• $n \geq 4$: Poisson–Lyapounov–Nekhoroshev (PLN) systems (rank $N-(n-2)$). The $2(N-n+1)$ nontrivial integrals are structured into two commuting families (Gubbiotti et al., 1 Dec 2025).

6. Comparison: $\mathfrak{g}_n$ versus $\mathfrak{sl}_n \ltimes \mathbb{C}^n$ and Symmetric Invariants

The alternative construction $g_n = \mathfrak{sl}_n \ltimes \mathbb{C}^n$, with $V$ as the standard representation, admits only the usual $\mathfrak{sl}_n$ Casimirs as symmetric invariants for $n \geq 4$. For $n=3$, an additional bi-homogeneous invariant $F(A, x) = \det(x, A^T x, (A^T)^2 x)$ exists, giving a polynomial ring of invariants matching the index. Invariant rings $S(g_n)^{g_n}$ are free in all ranks:

• $n=2$: generated by quadratic $\mathrm{Tr}(A^2)$,
• $n=3$: by $\mathrm{Tr}(A^2)$, $\mathrm{Tr}(A^3)$, and determinant $F(A,x)$,
• $n \geq 4$: by $\mathrm{Tr}(A^k)$, $2 \leq k \leq n$ only, no mixed vector invariants survive (Yakimova, 2015).

7. Applications and Mathematical Significance

The $\mathfrak{g}_n$ chain yields a hierarchy of Lie algebras central to the algebraic construction of integrable and quasi-integrable Hamiltonian systems. The unique Casimir structure and the Poisson coalgebra symmetry facilitate the explicit generation of polynomial integrals, underpinning superintegrability and PLN-type phenomena in classical mechanics. These results generalize and unify integrable systems associated with lower-rank Lie algebras, extending to higher-dimensional structures with distinctive properties. The full algebraic, coalgebraic, and invariant-theoretic classifications provided for both $\mathfrak{g}_n$ and $\mathfrak{sl}_n \ltimes \mathbb{C}^n$ form a comprehensive foundation for further applications in mathematical physics and representation theory (Gubbiotti et al., 1 Dec 2025, Yakimova, 2015).