[Paper Review] Les invariants polynômes de la représentation coadjointe de groupes inhomogènes

Mustapha Raïs|ArXiv.org|Mar 30, 2009

Advanced Algebra and Geometry3 references3 citations

TL;DR

This paper explicitly constructs polynomial invariants for the coadjoint representation of inhomogeneous Lie groups, including $ISL(n)$, $IO(n)$, $ISO(n)$, and $GL(n)\times(V\oplus V^*)$, by identifying generators of the invariant algebras via trace-based symmetric polynomials and Pfaffians. The key result is a complete system of algebraically independent generators for these invariants, with explicit formulas derived from characteristic polynomials and their gradients, extending earlier work on semi-direct products of Lie algebras.

ABSTRACT

Explicit generators are given for the ring of invariant polynomials under the coadjoint representation of certain inhomogeneous groups.

Motivation & Objective

• Address the structure of polynomial invariants in the coadjoint representation of inhomogeneous Lie groups, particularly semi-direct products $\mathfrak{g} \times V$.
• Provide explicit generators for the invariant algebras $Y(\mathfrak{p})$ in $S(\mathfrak{p})$ for $\mathfrak{p} = \mathfrak{g} \times V$ with $\mathfrak{g} = \mathfrak{sl}(n)$, $\mathfrak{o}(n)$, or $\mathfrak{so}(n)$.
• Extend previous results on $ISL(n)$ and $IO(n)$ by giving explicit formulas for invariants in terms of matrix traces and characteristic polynomials.
• Establish a connection between the invariants and the Pfaffian in odd-dimensional cases, particularly for $ISO(n)$ with $n=2\ell+1$.
• Clarify the relationship between the invariant algebras of $IO(n)$ and $ISO(n)$, showing that the latter requires an 'exotic' invariant not in the image of the standard trace construction.

Proposed method

• The paper uses the Duflo isomorphism to reduce the study of the center of the universal enveloping algebra to the algebra of polynomial invariants in the symmetric algebra $S(\mathfrak{p})$.
• Generators are constructed via the gradients $B_k(x)$ of the coefficients of the characteristic polynomial of $x \in \mathfrak{gl}(n)$, computed using the trace form.
• Polynomial invariants are defined as determinants of matrices formed from $v^* B_k(y)$, yielding a single generator $f$ for $ISL(n)$, and extended to multiple generators for $IO(n)$ and $ISO(n)$.
• Explicit formulas for invariants are derived using the matrix $Y = \begin{pmatrix} y & -{}^t w^* \\ w^* & 0 \end{pmatrix}$, whose Pfaffian gives the key invariant $\Phi$ in the odd-dimensional case.
• The method relies on restricting invariants to a specific submanifold $\mathfrak{h} \subset \mathfrak{p}^*$, where the invariants are shown to generate a polynomial algebra.
• Techniques from representation theory, including $Ad^*$-invariance and covariant polynomials $\Phi_k(y,v^*) = v^* B_k(y)$, are used to prove dimensionality and independence of the generating set.

Experimental results

Research questions

• RQ1 What is a complete system of algebraically independent polynomial generators for the coadjoint invariants of $ISL(n)$?
• RQ2 How do the invariants of $IO(n)$ and $ISO(n)$ differ, particularly in the odd-dimensional case?
• RQ3 What is the role of the Pfaffian in constructing an 'exotic' invariant for $ISO(n)$ when $n=2\ell+1$?
• RQ4 How do the invariants of $GL(n)\times(V\oplus V^*)$ relate to those of $IO(n)$ and $ISO(n)$?
• RQ5 What is the structure of the invariant algebra $Y(\mathfrak{p})$ for $\mathfrak{p} = \mathfrak{g} \times V$ with $\mathfrak{g} = \mathfrak{so}(n)$, and how does it relate to the $\mathbb{Z}_2$-contraction framework?

Key findings

• For $ISL(n)$, the algebra of invariants $Y(\mathfrak{p})$ is generated by the restriction $\overline{f}$ of a determinant function $f(y,v^*) = \det(v^*B_{n-1}(y), \dots, v^*)$, which is $SL(n)$-covariant and transforms by $\det(g)^{-1}$ under the coadjoint action.
• In the case $IO(n)$ with $n=2\ell+1$, the invariant algebra $J$ is generated by $\psi_0, \dots, \psi_\ell$, where $\psi_k(y,w^*) = p_{2k+2}(Y) - p_{2k+2}(y)$ for $Y = \begin{pmatrix} y & -{}^t w^* \\ w^* & 0 \end{pmatrix}$, and these are algebraically independent.
• For $ISO(n)$ with $n=2\ell+1$, the invariant algebra $Y(\mathfrak{c})$ is generated by $\psi_0, \dots, \psi_{\ell-1}$ and an 'exotic' invariant $\Phi$ satisfying $\Phi^2 = \psi_\ell$, with $\Phi(y,w^*) = w^* \cdot pf(y)$, where $pf(y)$ is the vector Pfaffian.
• The restriction of the invariant $\overline{f}$ to a specific subalgebra $\mathfrak{h}$ yields a polynomial $t$ that matches Kaneta's generator, confirming consistency with earlier work.
• An explicit isomorphism is established between the invariant algebra $Y(\mathfrak{p})$ and $\mathbb{C}[\varphi_0, \dots, \varphi_\ell]$, where $\varphi_k$ are symmetric polynomials in the eigenvalues of $y$ and $w^*$, showing algebraic independence.
• The algebra $J$ of invariants for $IO(n)$ is isomorphic to a quotient of $Y(\mathfrak{b})$ for $\mathfrak{b} = \mathfrak{gl}(n) \times (V \oplus V^*)$, suggesting a deeper structural relationship between the invariants of the larger and smaller groups.