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The Lie group denoted (Alekseevskii 68, Gray 69) or just is the quotient group of the direct product group of the given quaternion unitary groups by their diagonal center cyclic group of order 2.
A smooth manifold of dimension with G-structure for this group is a quaternion-Kähler manifold.
Similarly, for , spin groups in some dimension, the group denoted or just is the quotient group of the direct product group by the diagonal center cyclic group of order 2.
These products are examples of central products of groups.
For with , the Lie group denoted or just is the quotient group of the direct product group of quaternion unitary groups (in particular Spin(3)) by the diagonal center cyclic group of order 2 :
hence the quotient group by the subgroup
(e.g. Čadek-Vanžura 97, Sec. 2)
A similar definition yields
Write
for the quotient group of the direct product group of spin groups by their diagonal subgroup
(McInnes 99a, p. 9, Hilgert-Neeb 12, Prop. 17.3.1)
Sometimes one sees the notation further generalized to include cases such as
The direct product group has a canonical action on the quaternion vector space , where the factor Sp(n) acts as quaternion unitary matrix multiplication from the left, and acts by diagonal matrix action on each -summand from the right.
For instance for this action controls the quaternionic Hopf fibration and its equivariance (see there).
But this action is not an effective group action: Precisely the diagonal center (1) acts trivially.
There is then a commuting diagram of Lie groups
with the horizontal maps being group homomorphisms to Spin(8) and SO(8), respectively, the left morphism being the defining quotient projection and the right morphism the double cover morphism that defines the spin group.
(e.g. Čadek-Vanžura 97, p. 4)
(Marchiafava-Romani 76, Salamon 82, around Def. 2.1)
(…)
The case of for is special, as in this case the canonical inclusion becomes an isomorphism
with the special orthogonal group SO(4), and hence the compatibility diagram (2) now exhibits at the top the exceptional isomorphism Spin(4) (see there)
In summary:
There is a commuting diagram of Lie groups of the form
where
in the top right we have Spin(4),
in the bottom left we have Sp(1).Sp(1)
in the bottom right we have SO(4)
the horizontal morphism assigns the conjugation action of unit quaternions, as indicated,
the right vertical morphism is the defining double cover,
the left vertical morphism is the defining quotient group-projection.
For , group in Def. is the group otherwise known as spin^c(n):
This is due to the identification of the double cover by Spin(2) of SO(2) with the real Hopf fibration (this Prop), which identifies compatible with the subgroupinclusion of .
(See also e.g. Gompf 97, p. 2)
(Spin(5).Spin(3)-subgroups in SO(8))
The direct product group SO(3) SO(5) together with the groups Sp(2).Sp(1) and , with their canonical inclusions into SO(8), form 3 conjugacy classes of subgroups inside SO(8), and the triality group acts transitively on these three classes.
Similarly:
(Spin(5).Spin(3)-subgroups in Spin(8))
The groups Spin(5).Spin(3), Sp(2).Sp(1) and , with their canonical inclusions into Spin(8), form 3 conjugacy classes of subgroups inside Spin(8), and the triality group acts transitively on these three classes.
In summary:
The group
is the quotient group of the direct product group of Spin(4) with Spin(3) by the subgroup
Due to the exception isomorphism Spin(4) Spin(3) Spin(3) (this Prop.) this is isomorphic to the quotient group of the direct product of 3 copies of Sp(1) Spin(3) with itself
by the triple diagonal center
See the references below.
The coset space of Sp(2).Sp(1) (Def. ) by Sp(1)Sp(1)Sp(1) (Def. ) is the 4-sphere:
This follows essentially from the quaternionic Hopf fibration and its -equivariance…
(e.g. Bettiol-Mendes 15, (3.1), (3.2), (3.3))
We have the following coset spaces of spin groups by dot-products of Spin groups as above:
is the space of Cayley 4-planes (Cayley 4-form-calibrated submanifolds in 8d Euclidean space). This happens to also be homeomorphic to just the plain Grassmannian of 4-planes in 7d (recalled e.g. in Ornea-Piccini 00, p. 1).
Similarly,
is the Grassmannian of those Cayley 4-planes that are also special Lagrangian submanifolds (BBMOOY 96, p. 7 (8 of 17)).
Moreover,
is the Grassmannian of 3-planes in 8d. (Cadek-Vanzura 97, Lemma 2.6).
rotation groups in low dimensions:
see also
Very early appearances of the notation are mostly in discussions of Berger's theorem for exceptional holonomy:
Alfred Gray, A Note on Manifolds Whose Holonomy Group is a Subgroup of Sp(n) Sp(1), Michigan Math. J. Volume 16, Issue 2 (1969), 125-128.
Dmitri Alekseevskii, Riemannian spaces with exceptional holonomy groups, Functional Analalysis and its Applications (1968) 2: 97.
However, the even earlier paper:
describes this construction as a “local direct product” of topological groups and applies it to the classification of quaternionic manifolds. The notation in the classical paper of Bonan for this group is .
Of early algebraic interest is the structure theory article:
More on the cohomology of and its classifying space:
Stefano Marchiafava, Giuliano Romani, Alcune osservazioni sui sottogruppi abeliani del gruppo , Annali di Matematica 1977 (doi:10.1007/BF02413792)
Paolo Piccinni, Giuliano Romani, A generalization of symplectic Pontrjagin classes to vector bundles with structure group , Annali di Matematica pura ed applicata (1983) 133: 1 (doi:10.1007/BF01766008)
Paolo Piccinni, Vector fields and characteristic numbers on hyperkàhler and quaternion Kâhler manifolds, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1992) Volume: 3, Issue: 4, page 295-298 (dml:244204)
Dmitri Alekseevskii S. Marchiafava, Quaternionic structures on a manifold and subordinated structures, Annali di Matematica pura ed applicata (1996) 171: 205 (doi:10.1007/BF01759388)
Discussion of the lift to appears in
S. Marchiafava, G. Romani, Sui fibrati con struttura quaternionale generalizzata, Ann. Mat. Pura Appl. (IV) CVII, 131-157 (1976) (doi:10.1007/BF02416470)
Simon Salamon, around Def. 2.1 in Quaternionic Kähler manifolds, Invent Math (1982) 67: 143. (doi:10.1007/BF01393378)
Articles dealing specifically with the group :
Martin Čadek, Jiří Vanžura, Section 2 of On and -structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)
Martin Čadek, Jiří Vanžura, Almost quaternionic structures on eight-manifolds, Osaka J. Math. Volume 35, Number 1 (1998), 165-190 (euclid:1200787905)
Andreas Kollross, Prop. 3.3 of A Classification of Hyperpolar and Cohomogeneity One Actions, Transactions of the American Mathematical Society Vol. 354, No. 2 (Feb., 2002), pp. 571-612 (jstor:2693761)
See also the references at quaternion-Kähler manifold.
A textbook occurrence of dot notation for general spin groups, , appears in
The identification of with Spin^c appears for instance in
Discussion of central product spin groups as subgroups of semi-spin groups (motivated by analysis of the gauge groups and Green-Schwarz anomaly cancellation of heterotic string theory) is in
Brett McInnes, p. 9 of The Semispin Groups in String Theory, J. Math. Phys. 40:4699-4712, 1999 (arXiv:hep-th/9906059)
Brett McInnes, Gauge Spinors and String Duality, Nucl. Phys. B577:439-460, 2000 (arXiv:hep-th/9910100)
As such these also appear as U-duality groups and their subgroups, e.g.
The group (Example ) is discussed in the following, largely in describing the Grassmannian of Cayley 4-planes, see there:
Wu-Chung Hsiang, Wu-Yi Hsiang, Tables A of Differentiable Actions of Compact Connected Classical Groups: II, Annals of Mathematics Second Series, Vol. 92, No. 2 (1970), pp. 189-223 (jstor:1970834)
Reese Harvey, H. Blaine Lawson, theorem 1.38 of Calibrated geometries, Acta Math. Volume 148 (1982), 47-157 (Euclid:1485890157)
Robert Bryant, Reese Harvey, (3.19) in Submanifolds in Hyper-Kähler Geometry, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 1-31 (jstor:1990911)
Herman Gluck, Dana Mackenzie, Frank Morgan, (5.20) in Volume-minimizing cycles in Grassmann manifolds, Duke Math. J. Volume 79, Number 2 (1995), 335-404 (euclid:1077285156)
Megan M. Kerr, Lemma 6.2 of Some New Homogeneous Einstein Metrics on Symmetric Spaces, Transactions of the American Mathematical Society, Vol. 348, No. 1 (1996), pp. 153-171 (jstor:2155169)
Katrin Becker, Melanie Becker, David Morrison, Hirosi Ooguri, Y. Oz, Z. Yin, (3.5) of Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds, Nucl. Phys. B480:225-238, 1996 (arXiv:hep-th/9608116)
Victor Kac, A.V. Smilga, around (1.10) in Vacuum structure in supersymmetric Yang-Mills theories with any gauge group, in The Many Faces of the Superworld, pp. 185-234 World Scientific (2000) (arXiv:hep-th/9902029, doi:10.1142/9789812793850_0014)
Liviu Ornea, Paolo Piccinni, Cayley 4-frames and a quaternion-Kähler reduction related to Spin(7), Proceedings of the International Congress of Differential Geometry in the memory of A. Gray, held in Bilbao, Sept. 2000 (arXiv:math/0106116)
Karsten Grove, Burkhard Wilking, Wolfgang Ziller, p. 30 of Positively Curved Cohomogeneity One Manifolds and 3-Sasakian Geometry (arXiv:math/0511464)
Renato G. Bettiol, Ricardo A. E. Mendes, Flag manifolds with strongly positive curvature, Math. Z. 280 (2015), no. 3-4, 1031-1046 (arXiv:1412.0039)
Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Prop. 2.2 in Clifford systems in octonionic geometry (arXiv:1511.06239)
Discussion of in the context of super Lie algebras and superconformal symmetry is in:
and possibly with the -quotient not made explicit:
Peter Goddard (auth.), Peter Freund, K. T. Mahanthappa, p. 128 of Superstrings, NATO ASI Series 175, Springer 1988
Kazuo Hosomichi, Sangmin Lee, Sungjay Lee, Jaemo Park, slide 13 of New Superconformal Chern-Simons Theories (pdf)
Last revised on September 17, 2023 at 11:09:34. See the history of this page for a list of all contributions to it.