Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Holonomy
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==Affine holonomy== Affine holonomy groups are the groups arising as holonomies of torsion-free [[affine connection]]s; those which are not Riemannian or pseudo-Riemannian holonomy groups are also known as non-metric holonomy groups. The de Rham decomposition theorem does not apply to affine holonomy groups, so a complete classification is out of reach. However, it is still natural to classify irreducible affine holonomies. On the way to his classification of Riemannian holonomy groups, Berger developed two criteria that must be satisfied by the Lie algebra of the holonomy group of a torsion-free affine connection which is not [[symmetric space|locally symmetric]]: one of them, known as ''Berger's first criterion'', is a consequence of the Ambrose–Singer theorem, that the curvature generates the holonomy algebra; the other, known as ''Berger's second criterion'', comes from the requirement that the connection should not be locally symmetric. Berger presented a list of groups acting irreducibly and satisfying these two criteria; this can be interpreted as a list of possibilities for irreducible affine holonomies. Berger's list was later shown to be incomplete: further examples were found by [[Robert Bryant (mathematician)|R. Bryant]] (1991) and by Q. Chi, S. Merkulov, and L. Schwachhöfer (1996). These are sometimes known as ''exotic holonomies''. The search for examples ultimately led to a complete classification of irreducible affine holonomies by Merkulov and Schwachhöfer (1999), with Bryant (2000) showing that every group on their list occurs as an affine holonomy group. The Merkulov–Schwachhöfer classification has been clarified considerably by a connection between the groups on the list and certain symmetric spaces, namely the [[hermitian symmetric space]]s and the [[quaternion-Kähler symmetric space]]s. The relationship is particularly clear in the case of complex affine holonomies, as demonstrated by Schwachhöfer (2001). Let ''V'' be a finite-dimensional complex vector space, let ''H'' ⊂ Aut(''V'') be an irreducible semisimple complex connected Lie subgroup and let ''K'' ⊂ ''H'' be a [[maximal compact subgroup]]. # If there is an irreducible hermitian symmetric space of the form ''G''/(U(1) · ''K''), then both ''H'' and '''C'''*· ''H'' are non-symmetric irreducible affine holonomy groups, where ''V'' the tangent representation of ''K''. # If there is an irreducible quaternion-Kähler symmetric space of the form ''G''/(Sp(1) · ''K''), then ''H'' is a non-symmetric irreducible affine holonomy groups, as is '''C'''* · ''H'' if dim ''V'' = 4. Here the complexified tangent representation of Sp(1) · ''K'' is '''C'''<sup>2</sup> ⊗ ''V'', and ''H'' preserves a complex symplectic form on ''V''. These two families yield all non-symmetric irreducible complex affine holonomy groups apart from the following: :<math> \begin{align} \mathrm{Sp}(2, \mathbf C) \cdot \mathrm{Sp}(2n, \mathbf C) &\subset \mathrm{Aut}\left(\mathbf C^{2} \otimes\mathbf C^{2n}\right)\\ G_2(\mathbf C) &\subset \mathrm{Aut}\left(\mathbf C^7\right)\\ \mathrm{Spin}(7, \mathbf C) &\subset \mathrm{Aut}\left(\mathbf C^8\right). \end{align}</math> Using the classification of hermitian symmetric spaces, the first family gives the following complex affine holonomy groups: :<math> \begin{align} Z_{\mathbf C} \cdot \mathrm{SL}(m, \mathbf C) \cdot \mathrm{SL}(n, \mathbf C) &\subset \mathrm{Aut}\left(\mathbf C^m\otimes\mathbf C^n\right)\\ Z_{\mathbf C} \cdot \mathrm{SL}(n, \mathbf C) &\subset \mathrm{Aut}\left(\Lambda^2\mathbf C^n\right)\\ Z_{\mathbf C} \cdot \mathrm{SL}(n, \mathbf C) &\subset \mathrm{Aut}\left(S^2\mathbf C^n\right)\\ Z_{\mathbf C} \cdot \mathrm{SO}(n, \mathbf C) &\subset \mathrm{Aut}\left(\mathbf C^n\right)\\ Z_{\mathbf C} \cdot \mathrm{Spin}(10, \mathbf C) &\subset \mathrm{Aut}\left(\Delta_{10}^+\right) \cong \mathrm{Aut}\left(\mathbf C^{16}\right)\\ Z_{\mathbf C} \cdot E_6(\mathbf C) &\subset \mathrm{Aut}\left(\mathbf C^{27}\right) \end{align}</math> where ''Z''<sub>'''C'''</sub> is either trivial, or the group '''C'''*. Using the classification of quaternion-Kähler symmetric spaces, the second family gives the following complex symplectic holonomy groups: :<math> \begin{align} \mathrm{Sp}(2, \mathbf C) \cdot \mathrm{SO}(n, \mathbf C) &\subset \mathrm{Aut}\left(\mathbf C^2\otimes\mathbf C^n\right)\\ (Z_{\mathbf C}\,\cdot)\, \mathrm{Sp}(2n, \mathbf C) &\subset \mathrm{Aut}\left(\mathbf C^{2n}\right)\\ Z_{\mathbf C} \cdot\mathrm{SL}(2, \mathbf C) &\subset \mathrm{Aut}\left(S^3\mathbf C^2\right)\\ \mathrm{Sp}(6, \mathbf C) &\subset \mathrm{Aut}\left(\Lambda^3_0\mathbf C^6\right)\cong \mathrm{Aut}\left(\mathbf C^{14}\right)\\ \mathrm{SL}(6, \mathbf C) &\subset \mathrm{Aut}\left(\Lambda^3\mathbf C^6\right)\\ \mathrm{Spin}(12, \mathbf C) &\subset \mathrm{Aut}\left(\Delta_{12}^+\right) \cong \mathrm{Aut}\left(\mathbf C^{32}\right)\\ E_7(\mathbf C) &\subset \mathrm{Aut}\left(\mathbf C^{56}\right)\\ \end{align}</math> (In the second row, ''Z''<sub>'''C'''</sub> must be trivial unless ''n'' = 2.) From these lists, an analogue of Simons's result that Riemannian holonomy groups act transitively on spheres may be observed: the complex holonomy representations are all [[prehomogeneous vector space]]s. A conceptual proof of this fact is not known. The classification of irreducible real affine holonomies can be obtained from a careful analysis, using the lists above and the fact that real affine holonomies complexify to complex ones.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)