[Paper Review] Homotopes of Symmetric Spaces II. Structure Variety and Classification

Wolfgang Bertram, Pierre Bieliavsky|arXiv (Cornell University)|Nov 13, 2010

Advanced Topics in Algebra16 references3 citations

TL;DR

This paper classifies homotopes of classical symmetric spaces by leveraging their fibered structure: homotopes are symmetric spaces fibered over non-degenerate bases with flat, split fibers. The key contribution is a complete parametrization of structure varieties using inner ideals in classical Jordan pairs and their complementation, with a notable exception for degenerate inner ideals in skew-symmetric matrices.

ABSTRACT

We classify homotopes of classical symmetric spaces (studied in Part I of this work). Our classification uses the fibered structure of homotopes: they are fibered as symmetric spaces, with flat fibers, over a non-degenerate base; the base spaces correspond to inner ideals in Jordan pairs. Using that inner ideals in classical Jordan pairs are always complemented (in the sense defined by O. Loos and E. Neher), the classification of homotopes is obtained by combining the classification of inner ideals with the one of isotopes of a given inner ideal.

Motivation & Objective

• To provide a complete classification of homotopes for classical symmetric spaces, extending results from Part I.
• To establish that regular homotopes admit a fibered structure over non-degenerate symmetric spaces with flat, split fibers.
• To clarify the role of inner ideals in the structure variety and their complementation in Jordan pairs.
• To resolve the parametrization of structure varieties by combining classification of inner ideals with isotopes of each ideal.
• To identify and exclude a special class of degenerate homotopes arising from 'point spaces' in skew-symmetric matrix spaces.

Proposed method

• Uses the structure variety of a Jordan pair $(V^+, V^-)$, defined as the set of linear maps $\alpha: V^+ \to V^-$ satisfying a trilinear compatibility condition.
• Defines homotopes via trilinear maps $T_\alpha(x,y,z) = T^+(x, \alpha y, z)$, which yield Lie triple systems when $T^\pm$ form a Jordan pair.
• Introduces regularity of $\alpha$ as equivalent to the image $I = \mathrm{Im}\,\alpha$ being a complemented inner ideal in $V^-$.
• Applies the complementation theory of Loos and Neher to ensure the short exact sequence of Lie triple systems splits.
• Combines classification of inner ideals in classical Jordan pairs with classification of isotopes (invertible elements in $\mathrm{Svar}(I)$) to reconstruct all homotopes.
• Analyzes special cases via real and complex Jordan algebras isomorphisms, including $\mathbb{R}^n \cong \mathbb{H}$, $M(2,2;\mathbb{R})$, $\mathrm{Herm}(2,\mathbb{C})$, and $M(1,4;\mathbb{R})$.

Experimental results

Research questions

• RQ1 Which homotopes of classical symmetric spaces are parametrized by the structure variety, and are the lists in Part I exhaustive?
• RQ2 How does the fibered structure of homotopes arise from the algebraic properties of inner ideals in Jordan pairs?
• RQ3 Under what conditions does the fibration of a homotope over its base space split, and what is the role of complementation?
• RQ4 Why do homotopes associated with 'point spaces' in $\mathrm{Asym}(n,\mathbb{K})$ fail to be captured by the construction in Part I?
• RQ5 What is the relationship between isotopes of an inner ideal and the classification of homotopes?

Key findings

• The classification of homotopes of classical symmetric spaces is complete, except for a special class of degenerate homotopes arising from 'point spaces' in $\mathrm{Asym}(n,\mathbb{K})$.
• All regular homotopes are fibered as symmetric spaces over non-degenerate symmetric base spaces with flat, split fibers.
• The regularity of a homotope $\alpha$ is equivalent to the image $I = \mathrm{Im}\,\alpha$ being a complemented inner ideal in the target Jordan pair.
• For simple real finite-dimensional Jordan pairs, every inner ideal is complemented, ensuring the fibered structure holds for all $\alpha$.
• The structure variety is fully parametrized by combining the classification of inner ideals in classical Jordan pairs with the classification of isotopes of each ideal.
• In low dimensions (e.g., 6), special isomorphisms (e.g., $\mathrm{Asym}(3,\mathbb{C}) \cong M(1,3;\mathbb{C})$) lead to non-generic symmetric space families with distinct contraction patterns.