nLab
Sp(n).Sp(1)

Contents

Contents

Idea

The Lie group denoted Sp(n).Sp(1)Sp(n).Sp(1) (Alekseevskii 68, Gray 69) or just Sp(n)Sp(1)Sp(n)Sp(1) 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 4n4n with G-structure for this group G=Sp(n).Sp(1)G = Sp(n).Sp(1) is a quaternion-Kähler manifold.

Similarly, for Spin(n 1)Spin(n_1), Spin(n 2)Spin(n_2) spin groups in some dimension, the group denoted Spin(n 1)Spin(n 2)Spin(n_1) \cdot Spin(n_2) or just Spin(n 1)Spin(n 2)Spin(n_1)Spin(n_2) is the quotient group of the direct product group Spin(n 1)×Spin(n 2)Spin(n_1) \times Spin(n_2) by the diagonal center cyclic group of order 2.

These products G 1G 2G_1 \cdot G_2 are examples of central products of groups.


Definition

Definition

For nn \in \mathbb{N} with n2n \geq 2, the Lie group denoted Sp(n).Sp(1)Sp(n).Sp(1) or just Sp(n)Sp(1)Sp(n)Sp(1) is the quotient group of the direct product group Sp(n)×Sp(1)Sp(n) \times Sp(1) of quaternion unitary groups Sp(n)Sp(n) (in particular Sp(1)Sp(1) \simeq Spin(3)) by the diagonal center cyclic group of order 2 2\mathbb{Z}_2:

Sp(n).Sp(1)(Sp(n)×Sp(1))/ diag 2 Sp(n).Sp(1) \;\coloneqq\; \big( Sp(n) \times Sp(1) \big)/_{diag}\mathbb{Z}_2

hence the quotient group by the subgroup

(1) 2{(1,1),(1,1)}Sp(n)×Sp(1). \mathbb{Z}_2 \;\simeq\; \big\{ (1,1), (-1,-1) \big\} \hookrightarrow Sp(n) \times Sp(1) \,.

(e.g. Čadek-Vanžura 97, Sec. 2)

A similar definition yields

Definition

Write

Spin(n 1)Spin(n 2)(Spin(n 1)×Spin(n 2))/ 2 Spin(n_1) \cdot Spin(n_2) \;\coloneqq\; \big( Spin(n_1) \times Spin(n_2) \big)/\mathbb{Z}_2

for the quotient group of the direct product group of spin groups by their diagonal subgroup

2{(1,1),(1,1)}Spin(n 1)×Spin(n 1). \mathbb{Z}_2 \;\simeq\; \big\{ (1,1), (-1,-1) \big\} \;\hookrightarrow\; Spin(n_1) \times Spin(n_1) \,.

(McInnes 99a, p. 9, Hilgert-Neeb 12, Prop. 17.3.1)

Sometimes one sees the notation further generalized to include cases such as

  • Spin(n)U(1)Spin(n)Spin(2)Spin(n) \cdot U(1) \simeq Spin(n)\cdot Spin(2) \simeq Spin^c,

see Example below.


Properties

As the effective quotient of Sp(n)×Sp(1)Sp(n)\times Sp(1) acting on n\mathbb{H}^n

The direct product group Sp(n)×Sp(1)Sp(n) \times Sp(1) has a canonical action on the quaternion vector space n\mathbb{H}^n, where the factor Sp(n) acts as 2×22 \times 2 quaternion unitary matrix multiplication from the left, and Sp(1)Sp(1) acts by diagonal 1×11 \times 1 matrix action on each \mathbb{H}-summand from the right.

For instance for n=2n = 2 this action controls the quaternionic Hopf fibration and its Sp(2)Sp(2) 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

(2)Sp(2)×Sp(1) Spin(8) Sp(2)Sp(1) SO(8) \array{ Sp(2) \times Sp(1) &\longrightarrow& Spin(8) \\ \big\downarrow && \big\downarrow \\ Sp(2) \cdot Sp(1) &\longrightarrow& SO(8) }

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)

Lift to Sp(n)×Sp(1)Sp(n) \times Sp(1)

(Marchiafava-Romani 76, Salamon 82, around Def. 2.1)

(…)


Examples

Sp(1)Sp(1)Sp(1)\cdot Sp(1) is SO(4)SO(4)

The case of Sp(n)Sp(1)Sp(n)\cdot Sp(1) for n=1n = 1 is special, as in this case the canonical inclusion Sp(n)Sp(1)SO(4n)Sp(n)\cdot Sp(1) \hookrightarrow SO(4n) becomes an isomorphism

Sp(1)Sp(1)SO(4) Sp(1)\cdot Sp(1) \;\simeq\; SO(4)

with the special orthogonal group SO(4), and hence the compatibility diagram (2) now exhibits at the top the exceptional isomorphism Sp(1)×Sp(1)Sp(1) \times Sp(1) \simeq Spin(4) (see there)

In summary:

Proposition

There is a commuting diagram of Lie groups of the form

(q 1,q 2) (xq 1xq¯ 2) Sp(1)×Sp(1) Spin(4) Sp(1)Sp(1) SO(4) \array{ ( q_1, q_2 ) &\mapsto& (x \mapsto q_1 \cdot x \cdot \overline{q}_2) \\ Sp(1) \times Sp(1) &\overset{\simeq}{\longrightarrow}& Spin(4) \\ \big\downarrow && \big\downarrow \\ Sp(1)\cdot Sp(1) &\overset{\simeq}{\longrightarrow}& SO(4) }

where

  1. in the top left we have Sp(1) = Spin(3),

  2. in the top right we have Spin(4),

  3. in the bottom left we have Sp(1).Sp(1)

  4. in the bottom right we have SO(4)

  5. the horizontal morphism assigns the conjugation action of unit quaternions, as indicated,

  6. the right vertical morphism is the defining double cover,

  7. the left vertical morphism is the defining quotient group-projection.


Spin(n)Spin(2)Spin(n)\cdot Spin(2) is Spin c(n)Spin^c(n)

Example

For nn \in \mathbb{N}, group Sp(n)Sp(2)Sp(n) \cdot Sp(2) in Def. is the group otherwise known as spin^c(n):

Spin(n)Spin(2)Spin c(n). Spin(n)\cdot Spin(2) \;\simeq\; 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 Spin(2)U(1)Spin(2) \simeq U(1) compatible with the subgroupinclusion of 2\mathbb{Z}_2.

(See also e.g. Gompf 97, p. 2)

Triality

Proposition

(Spin(5).Spin(3)-subgroups in SO(8))

The direct product group SO(3) ×\times SO(5) together with the groups Sp(2).Sp(1) and Sp(1)Sp(2)Sp(1) \cdot Sp(2), with their canonical inclusions into SO(8), form 3 conjugacy classes of subgroups inside SO(8), and the triality group Out(Spin(8))Out(Spin(8)) acts transitively on these three classes.

(Kollross 02, Prop. 3.3 (3))

Similarly:

Proposition

(Spin(5).Spin(3)-subgroups in Spin(8))

The groups Spin(5).Spin(3), Sp(2).Sp(1) and Sp(1)Sp(2)Sp(1) \cdot Sp(2), with their canonical inclusions into Spin(8), form 3 conjugacy classes of subgroups inside Spin(8), and the triality group Out(Spin(8))Out(Spin(8)) acts transitively on these three classes.

(Čadek-Vanžura 97, Sec. 2)

In summary:

Sp(1)Sp(1)Sp(1)=Spin(4)Spin(3)Sp(1)Sp(1)Sp(1) = Spin(4)\cdot Spin(3)

Example

(Spin(4).Spin(3))

The group

Spin(4)Spin(3)(Spin(4)×Spin(3))/ 2 Spin(4)\cdot Spin(3) \;\coloneqq\; \big( Spin(4) \times Spin(3) \big)/\mathbb{Z}_2

is the quotient group of the direct product group of Spin(4) with Spin(3) by the subgroup

(3) 2{(1,1),(1,1)}Spin(4)×Spin(3). \mathbb{Z}_2 \;\simeq\; \big\{ (1,1), (-1,-1) \big\} \hookrightarrow Spin(4) \times Spin(3) \,.

Due to the exception isomorphism Spin(4) \simeq Spin(3) ×\times Spin(3) (this Prop.) this is isomorphic to the quotient group of the direct product of 3 copies of Sp(1) \simeq Spin(3) with itself

Spin(4)Spin(3)Sp(1)Sp(1)Sp(1)(Spin(3)×Spin(3)×Spin(3))/ diag 2 Spin(4)\cdot Spin(3) \;\simeq\; Sp(1)Sp(1)Sp(1) \;\coloneqq\; \big( Spin(3) \times Spin(3) \times Spin(3)\big)/_{diag} \mathbb{Z}_2

by the triple diagonal center

(4) 2{(1,1,1),(1,1,1)}Spin(3)×Spin(3)×Spin(3). \mathbb{Z}_2 \;\simeq\; \big\{ (1,1,1), (-1,-1,-1) \big\} \hookrightarrow Spin(3) \times Spin(3) \times Spin(3) \,.

See the references below.

Example

The coset space of Sp(2).Sp(1) (Def. ) by Sp(1)Sp(1)Sp(1) (Def. ) is the 4-sphere:

Sp(2)Sp(1)Sp(1)Sp(1)Sp(1)S 4. \frac{ Sp(2)\cdot Sp(1) } { Sp(1)Sp(1)Sp(1) } \;\simeq\; S^4 \,.

This follows essentially from the quaternionic Hopf fibration and its Sp(2)Sp(2)-equivariance

(e.g. Bettiol-Mendes 15, (3.1), (3.2), (3.3))

Spin-Grassmannians

We have the following coset spaces of spin groups by dot-products of Spin groups as above:

Spin(7)/(Spin(4)Spin(3))SO(7)/(SO(4)×SO(3))Gr(4,7) Spin(7)/ \big( Spin(4)\cdot Spin(3) \big) \;\simeq\; SO(7) / \big( SO(4) \times SO(3) \big) \;\simeq\; Gr(4, 7)

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,

Spin(6)/(Spin(3)Spin(3))SU(6)/SO(4) Spin(6)/ \big( Spin(3)\cdot Spin(3) \big) \;\simeq\; SU(6)/ SO(4)

is the Grassmannian of those Cayley 4-planes that are also special Lagrangian submanifolds (BBMOOY 96, p. 7 (8 of 17)).

Moreover,

Spin(8)/(Spin(5)Spin(3))Gr(3,8) Spin(8)/ \big( Spin(5)\cdot Spin(3) \big) \;\simeq\; Gr(3, 8)

is the Grassmannian of 3-planes in 8d. (Cadek-Vanzura 97, Lemma 2.6).


rotation groups in low dimensions:

sp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)Pin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)SO(8)
SO(9)Spin(9)
\vdots\vdots
SO(16)Spin(16)SemiSpin(16)
SO(32)Spin(32)SemiSpin(32)

see also

References

Sp(n)Sp(1)Sp(n)\cdot Sp(1)

Very early appearances of the notation Sp(n)Sp(1)Sp(n)\cdot Sp(1) are mostly in discussions of Berger's theorem for exceptional holonomy:

However, the even earlier paper:

  • Joseph Wolff, Complex homogeneous contact manifolds and quaternionic symmetric spaces, Journal of Mathematics and Mechanics, vol. 14 (1965), pp. 1033-1048.

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 V 4n[Sp(n) HSp(1)]V_{4n} [Sp(n) \otimes_\mathbf{H} Sp(1)].

Of early algebraic interest is the structure theory article:

  • Stefano Marchiafava, Giuliano Romani, Sul classificante del gruppo Sp(n)Sp(1)Sp(n) \cdot Sp(1), Annali di Matematica Pura ed Applicata December 1976, Volume 110, Issue 1, pp 295–319 (doi:10.1007/BF02418010)

More on the cohomology of Sp(n)Sp(1)Sp(n)\cdot Sp(1) and its classifying space:

  • Stefano Marchiafava, Giuliano Romani, Alcune osservazioni sui sottogruppi abeliani del gruppo Sp(n)Sp(1)Sp(n)\cdot Sp(1), Annali di Matematica 1977 (doi:10.1007/BF02413792)

  • Paolo Piccinni, Giuliano Romani, A generalization of symplectic Pontrjagin classes to vector bundles with structure group Sp(n)Sp(1)Sp(n)\cdot Sp(1), 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 Sp(n)×Sp(1)Sp(n) \times Sp(1) appears in

Sp(2)Sp(1)Sp(2)\cdot Sp(1)

Articles dealing specifically with the group Sp(2)Sp(1)Sp(2)\cdot Sp(1):

  • Martin Čadek, Jiří Vanžura, Section 2 of On Sp(2)Sp(2) and Sp(2)Sp(1)Sp(2) \cdot Sp(1)-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.

Spin(n 1)Spin(n 2)Spin(n_1)\cdot Spin(n_2)

A textbook occurrence of dot notation for general spin groups, Spin(n 1)Spin(n 2)Spin(n_1)\cdot Spin(n_2), appears in

The identification of SpinSpin˙(2)Spin \dot Spin(2) 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

As such these also appear as U-duality groups and their subgroups, e.g.

Sp(1)Sp(1)Sp(1)Spin(4)Spin(3)Sp(1)Sp(1)Sp(1) \simeq Spin(4)\cdot Spin(3)

The group Spin(4)Spin(3)(Spin(3)) 3/ diag 2Spin(4)\cdot Spin(3) \simeq (Spin(3))^3/_{diag} \mathbb{Z}_2 (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 Sp(1)Sp(1)Sp(1)Sp(1)\cdot Sp(1) \cdot Sp(1) in the context of super Lie algebras and superconformal symmetry is in:

  • Peter Freund, p. 634 of World topology and gauged internal symmetries, Proc. 19th Int. Conf. High Energy Physics, Tokyo 1978 (spire:137780, pdf)

and possibly with the 2\mathbb{Z}_2-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)

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