[Paper Review] On the Cayley degree of an algebraic group

Nicole Lemire, Vladimir L. Popov+1 more|arXiv (Cornell University)|Aug 18, 2006

Algebraic Geometry and Number Theory2 references3 citations

TL;DR

This paper introduces the Cayley degree as a measure of how far a connected linear algebraic group is from being Cayley, generalizing the classical Cayley map. It establishes upper bounds for the Cayley degrees of special linear groups $\mathrm{SL}_n$ and the exceptional group $\mathrm{G}_2$, proving $\mathrm{Cay}(\mathrm{SL}_n) \leq n-2$ and $\mathrm{Cay}(\mathrm{G}_2) = 2$, using birational geometry and Weyl group actions on maximal tori and Lie algebras.

ABSTRACT

A connected linear algebraic group G is called a Cayley group if the Lie algebra of G endowed with the adjoint G-action and the group variety of G endowed with the conjugation G-action are birationally G-isomorphic. In particular, the classical Cayley map, X \mapsto (I_n-X)/(I_n+X), between the special orthogonal group SO_n and its Lie algebra so_n, shows that SO_n is a Cayley group. In an earlier paper (see math.AG/0409004) we classified the simple Cayley groups defined over an algebraically closed field of characteristic zero. Here we consider a new numerical invariant of G, the Cayley degree, which "measures" how far G is from being Cayley, and prove upper bounds on Cayley degrees of some groups.

Motivation & Objective

• To define and study the Cayley degree as a numerical invariant measuring the failure of a linear algebraic group to be Cayley.
• To extend the classification of Cayley groups beyond the known case of degree 1 to higher degrees.
• To establish upper bounds on the Cayley degree for reductive groups, particularly $\mathrm{SL}_n$ and $\mathrm{G}_2$, using rational equivariant maps.
• To connect the Cayley degree to representation theory via Weyl group actions on function fields of maximal tori.

Proposed method

• Define the Cayley degree $\mathrm{Cay}(G)$ as the minimal degree of a dominant, $G$-equivariant rational map from $G$ to its Lie algebra $\mathfrak{g}$, generalizing the classical Cayley map.
• Use a birational isomorphism $\psi$ between a maximal torus $T$ and a hypersurface $X$ in $\mathbb{A}^n$, then compose with a projection to the Lie algebra $\mathfrak{t}$ to construct generalized Cayley maps.
• Leverage the Weyl group $W = N_G(T)/T$ to reduce the problem to finite group actions on the field $k(T)$, enabling algebraic control over the degree.
• Construct explicit rational maps via symmetric functions and rational functions on $T$, such as $z_i = t_i - t_i^{-1}$, to generate subfields $k(M) \subset k(T)$ of controlled degree.
• Apply field extension degree $[k(T):k(M)]$ to bound the degree of the resulting generalized Cayley map, using the fact that $\deg \varphi = [k(G):\varphi^*(k(\mathfrak{g}))]$.
• Use the isomorphism between $\mathfrak{t}^*$ and a submodule $M$ of $k(T)$ to relate the map degree to the transcendence degree and field extensions.

Experimental results

Research questions

• RQ1 What is the minimal degree of a $G$-equivariant rational map from a reductive group $G$ to its Lie algebra, and how does it vary across different groups?
• RQ2 How can the Cayley degree be computed or bounded for groups like $\mathrm{SL}_n$ and $\mathrm{G}_2$ that are not Cayley?
• RQ3 To what extent can the Weyl group action on the function field of a maximal torus be used to construct and analyze generalized Cayley maps?
• RQ4 Can representation-theoretic techniques involving $W$-modules in $k(T)$ yield effective upper bounds on the Cayley degree?
• RQ5 Why does $\mathrm{G}_2$ have Cayley degree 2 despite not being Cayley, and how does this relate to its stable Cayley property?

Key findings

• The Cayley degree of $\mathrm{SL}_n$ is at most $n-2$, with equality for $n=4$, implying $\mathrm{Cay}(\mathrm{SL}_4) = 2$.
• The Cayley degree of the exceptional group $\mathrm{G}_2$ is exactly 2, showing it is close to being Cayley despite not being Cayley.
• For $n=3$, the new proof in the paper gives a simpler derivation of $\mathrm{Cay}(\mathrm{SL}_3) = 1$, confirming $\mathrm{SL}_3$ is Cayley.
• The construction of generalized Cayley maps relies on birational maps from a maximal torus to a hypersurface, followed by linear projections to the Lie algebra.
• The degree of the map is controlled by the field extension degree $[k(T):k(M)]$, where $M$ is a $W$-invariant submodule of $k(T)$ isomorphic to $\mathfrak{t}^*$.
• A representation-theoretic approach via $W$-modules in $k(T)$ yields an upper bound of 6 for $\mathrm{Cay}(\mathrm{G}_2)$, though the sharp bound is 2.