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The special orthogonal group or rotation group, denoted , is the group of rotations in a Cartesian space of dimension .
This is one of the classical Lie groups. It is the connected component of the neutral element in the orthogonal group .
For instance for we have the circle group.
It is the first step in the Whitehead tower of
the next step of which is the spin group.
In physics the rotation group is related to angular momentum.
On the ordinary cohomology of the classifying spaces and
with coefficients:
generated by the Stiefel-Whitney classes
with integer coefficients (integral cohomology):
generated by the integral Stiefel-Whitney classes and the Pontrjagin classes
(e.g. Brown 82, Feshbach 83, Pittie 91, Rudolph-Schmidt 17, Theorem 4.2.23)
ADE classification and McKay correspondence
For an -dimensional manifold a lift of the classifying map of the -principal bundle to which the tangent bundle is associated is the same as a choice of orientation of .
For the almost degenerate case there are exceptional isomorphisms of Lie groups
with the circle group and spin group in dimension 2.
the case of SO(8) is special, since in the ADE classification of simple Lie groups it corresponds to D4, which makes its representation theory enjoy triality.
rotation groups in low dimensions:
see also
Fivebrane group string group spin group special orthogonal group orthogonal group.
For general references see also at orthogonal group.
ordinary cohomology of the classifying spaces:
John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) [ISBN:9780691081229, doi:10.1515/9781400881826, pdf]
Edgar H. Brown, The Cohomology of and with Integer Coefficients, Proceedings of the American Mathematical Society Vol. 85, No. 2 (Jun., 1982), pp. 283-288 (jstor:2044298)
Mark Feshbach, The Integral Cohomology Rings of the Classifying Spaces of and , Indiana Univ. Math. J. 32 (1983), 511-516 (doi:10.1512/iumj.1983.32.32036)
Harsh Pittie, The integral homology and cohomology rings of SO(n) and Spin(n), Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (doi:10.1016/0022-4049(91)90108-E)
Howard Georgi, §21 & 22 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]
with an eye towards application to spinors in (the standard model of) particle physics
Gerd Rudolph, Matthias Schmidt, around Theorem 4.2.23 of: Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)
See also
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