Contents

group theory

# Contents

## Idea

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

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

These products $G_1 \cdot G_2$ are examples of central products of groups.

## Definition

###### Definition

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

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

hence the quotient group by the subgroup

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

A similar definition yields

###### Definition

Write

$Spin(n_1) \cdot Spin(n_2) \;\coloneqq\; \big( Spin(n_1) \times Spin(n_2) \big)/\mathbb{Z}_2$

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

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

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

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

see Example below.

## Properties

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

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

For instance for $n = 2$ this action controls the quaternionic Hopf fibration and its $Sp(2)$ 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)$\array{ Sp(2) \times Sp(1) &\longrightarrow& Spin(8) \\ \big\downarrow && \big\downarrow \\ Sp(2) \cdot Sp(1) &\longrightarrow& SO(8) }$

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.

(…)

## Examples

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

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

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

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

In summary:

###### Proposition

There is a commuting diagram of Lie groups of the form

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

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.

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

###### Example

For $n \in \mathbb{N}$, group $Sp(n) \cdot Sp(2)$ in Def. is the group otherwise known as 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 $Spin(2) \simeq U(1)$ compatible with the subgroupinclusion of $\mathbb{Z}_2$.

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

Similarly:

###### Proposition

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

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

In summary:

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

###### Example

(Spin(4).Spin(3))

The group

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

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

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

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

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

by the triple diagonal center

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

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:

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

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

### Spin-Grassmannians

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

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

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,

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

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

Moreover,

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

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

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)

## References

### $Sp(n)\cdot Sp(1)$

Very early appearances of the notation $Sp(n)\cdot Sp(1)$ 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 $V_{4n} [Sp(n) \otimes_\mathbf{H} Sp(1)]$.

Of early algebraic interest is the structure theory article:

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

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

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

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

### $Sp(2)\cdot Sp(1)$

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

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

### $Spin(n_1)\cdot Spin(n_2)$

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

The identification of $Spin \dot Spin(2)$ with Spin^c appears for instance in

• Robert Gompf, $Spin^c$ structures and homotopy equivalences, Geom. Topol. 1 (1997) 41-50 (arXiv:math/9705218)

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.

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

The group $Spin(4)\cdot Spin(3) \simeq (Spin(3))^3/_{diag} \mathbb{Z}_2$ (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 (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 $Sp(1)\cdot Sp(1) \cdot Sp(1)$ 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 $\mathbb{Z}_2$-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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