The generalized Cayley map from an algebraic group to its Lie algebra
| dc.creator | Kostant, Bertram | |
| dc.creator | Michor, Peter W. | |
| dc.date | 2001-09-10 | |
| dc.date | 2003-03-25 | |
| dc.date.accessioned | 2026-07-07T04:43:20Z | |
| dc.date.available | 2026-07-07T04:43:20Z | |
| dc.description | Each infinitesimally faithful representation of a reductive complex connected algebraic group $G$ induces a dominant morphism $Φ$ from the group to its Lie algebra $\g$ by orthogonal projection in the endomorphism ring of the representation space. The map $Φ$ identifies the field $Q(G)$ of rational functions on $G$ with an algebraic extension of the field $Q(\g)$ of rational functions on $\g$. For the spin representation of $\on{Spin}(V)$ the map $Φ$ essentially coincides with the classical Cayley transform. In general, properties of $Φ$ are established and these properties are applied to deal with a separation of variables (Richardson) problem for reductive algebraic groups: Find $\on{Harm}(G)$ so that for the coordinate ring $A(G)$ of $G$ we have $A(G) = A(G)^G\otimes \on{Harm}(G)$. As a consequence of a partial solution to this problem and a complete solution for SL(n) one has in general the equality $[Q(G):Q(\g)] = [Q(G)^G:Q(\g)^G]$ of the degrees of extension fields. Among other results, $Φ$ yields (for the complex case) a generalization, involving generic regular orbits, of the result of Richardson showing that the Cayley map, when $G$ is semisimple, defines an isomorphism from the variety of unipotent elements in $G$ to the variety of nilpotent elements in $\g$. In addition if $G$ is semisimple the Cayley map establishes a diffeomorphism between the real submanifold of hyperbolic elements in $G$ and the space of infinitesimal hyperbolic elements in $\g$. Some examples are computed in detail. | |
| dc.description | AmSTeX, 33 pages, some misprints corrected | |
| dc.identifier | https://arxiv.org/abs/math/0109066 | |
| dc.identifier | http://arxiv.org/abs/math/0109066 | |
| dc.identifier | In: The orbit method in geometry and physics: In honor of A. A. Kirillov. Eds. Duval, Guieu, Ovsienko. Progress in Mathematics 213, Birkhaeuser Verlag, 2003. 259-296 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62174 | |
| dc.subject | Representation Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 22E50 | |
| dc.title | The generalized Cayley map from an algebraic group to its Lie algebra | |
| dc.type | text |