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am giving this group its own entry, so that every entry using it (such as quaternion-Kähler manifold) makes sure that the notation is not being confused with the direct product group.
What’s the source of this notation?
added this statement:
There is then a commuting diagram of Lie groups
$\array{ Sp(2) \times Sp(1) &\longrightarrow& Spin(8) \\ \big\downarrow && \big\downarrow \\ Sp(2) \cdot Sp(1) &\longrightarrow& SO(8) }$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.
I suppose it’s true for general $n$, with 8 on the right generalized to $4n$, but I leave it as is for the moment.
added under Examples the following quick remark (deserves to be beautified):
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 (eq:CompatibilityDiagram) now exhibits at the top the exceptional isomorphism $Sp(1) \times Sp(1) \simeq$ Spin(4) (see there)
$\array{ Sp(1) \times Sp(1) &\overset{\simeq}{\longrightarrow}& Spin(4) \\ \big\downarrow && \big\downarrow \\ Sp(1) \cdot Sp(1) &\overset{\simeq}{\longrightarrow}& SO(4) }$Currently I’m traveling and in areas with very unreliable internet; soon I will be able to add something about this notation and fill out a few points of this quaternionic stuff. I seem to recall the origins of this notation are Italian and may come from Marchiafava in the early 70s, when hyperkähler research started really expanding.
Thanks! No rush, but when you do have time, I’d be interested in seeing this.
Does anyone discuss $Sp(1)Sp(1)Sp(1)$-structure on 8-manifolds?
Not sure if this is established notation: By $Sp(1)Sp(1)Sp(1)$ I mean the quotient group of $Sp(1)\times Sp(1) \times Sp(1)$ by the triply diagonal center $\mathbb{Z}/2 = \{(1,1,1), (-1,-1,-1)\}$. So that’s the group $Spin(4)\cdot Spin(3)$.
In other words, I am thinking of the subgroup
$Sp(1)Sp(1)Sp(1) \hookrightarrow Sp(2)\cdot Sp(1)$which is the canonical inclusion $Sp(1) \times Sp(1) \hookrightarrow Sp(2)$ “dotted” with the identity on $Sp(1)$.
What I’d really like to know is this:
the integral cohomology of $B \big( Sp(1) \times Sp(1)\big) \simeq B Spin(4)$ has a generator $\tfrac{1}{2}\chi + \tfrac{1}{4} p_1$. Might that generator extend to $B \big(Sp(1)Sp(1)Sp(1) \big)$?
[edit: the group in question is that in Lemma 6.2 of Kerr 96 ]
It is actually hard to pin down the earliest appearance of this notation in the literature. A very early case is Alfred Gray’s article A Note on Manifolds Whose Holonomy Group is a Subgroup of Sp(n) $\cdot$ Sp(1), and Alekseevskii from the same year in Riemannian spaces with exceptional holonomy groups, both the year immediately after Bonan presented (almost) quaternionic manifolds as G-structures in his classical 1967 article, where his notation is completely different. The Bonan notation is $Sp(n) \otimes_\mathbf{H} Sp(1)$. I will add these missing references to the page. Of more purely algebraic interest is Marchiafava and Romani’s Sul classificante del gruppo Sp(n) $\cdot$ Sp(1).
Exccellent, thanks! I have added pointers to your new references from the first lines of the main text.
I have added hyperlinks for authors: Dmitry Vladimirovich Alekseevsky and Alfred Gray
started an Examples-subsection “Spin-Grassmannians”. Currently I have this, to be polished and expanded:
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), which in turn is homeomorphic to just the plain Grassmannian of 4-planes in 7d (recalled e.g. in Ornea-Piccini 00, p. 1)
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).
added one more case to the section of “Spin Grassmannians”:
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. 8).
James, might you know a source that gives $H^4\Big( B \big(Sp(1)Sp(1)Sp(1)\big), \mathbb{Z}\Big)$? We are struggling with a factor of 1/2 in there…
added these pointers:
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)
added also pointer to these here:
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)
@Urs #15
surely there is a general method to take a simply-connected connected compact Lie group $G$ with central subgroup $\mathbb{Z}/2$, such that $H^4(B G,\mathbb{Z})$ is known, and calculate $H^4(B[G/(\mathbb{Z}/2)],\mathbb{Z})$?
Surely! ;-)
But never mind, the factor came to me under the shower.
[ removed ]
[ also removed, sorry for the noise]
@Urs did you manage to resolve it?
added pointer to Pro. 17.3.1 of
for an example of usage of the dot-notation applied to general Spin-groups
added pointer to
and used that for a new section Examples - Triality. (Same material now also at Spin(8) in the section “Subgroup lattice”)
Does anyone know the following:
The group $Sp(1)\cdot Sp(1) \cdot Sp(1)$ has an evident action of the symetric group $\Sigma_3$ by automorphisms permuting the three dot factors.
This $\Sigma_3$-action is probably related to triality?! How?
I am guessing as follows:
Probably there are inclusions of $Sp(1)\cdot Sp(1) \cdot Sp(1)$ into each of
$Sp(1)\cdot Sp(2)$
$Sp(2)\cdot Sp(1)$
$Spin(3)\cdot Spin(5)$
(these now all understood under their canonical embedding as subgroups of $Spin(8)$ as here) such that as these three get permuted into each other under the action of triality, their common subgroup $Sp(1)\cdot Sp(1) \cdot Sp(1)$ is fixed up to isomorphism, and these fixing isomorphisms are the $\Sigma_3$ action from before!?
Is this discussed anywhere?
I am thinking:
Since the subgroup isomorphism on the left of
$\array{ Sp(2)Sp(1) &\hookrightarrow& Spin(8) \\ \mathllap{\simeq}\big\downarrow && \big\downarrow\mathrlap{=} \\ Sp(1)Sp(2) &\hookrightarrow& Spin(8) }$manifestly comes from exchanging factors, and since the subgroup isomorphism on the left of
$\array{ Sp(2)Sp(1) &\hookrightarrow& Spin(8) \\ \mathllap{\simeq}\big\downarrow && \big\downarrow\mathrlap{=} \\ Spin(5)Spin(3) &\hookrightarrow& Spin(8) }$is the dot-product of the isomorphisms $Sp(2) \overset{\simeq}{\to} Spin(5)$ with $Sp(1) \overset{\simeq}{\to} Spin(3)$ (by the proof of Lemma 2.4 here), the proof of my conjecture (that the first and second inner circles here commute) is reduced to checking that the diagram
$\array{ Sp(1)Sp(1) &\hookrightarrow& Sp(2) \\ \mathllap{\simeq}\big\downarrow && \big\downarrow\mathrlap{\simeq} \\ Spin(3)Spin(3) &\hookrightarrow& Spin(5) }$commutes, where the iso on the left is either the dot-product of the iso $Sp(1) \stackrel{\simeq}{\to} Spin(3)$ with itself, or that followed by switching dot-factors.
If there is any justice in the world, then it does. But I’d need to dig deeper into the details to prove this.
cross-linked with central product of groups
there is a curious appearance of $Sp(1)\cdot Sp(1) \cdot Sp(1)$ in
Have added the pointer. Also I suspect the following two really mean $SU(1)\cdot SU(2) \cdot SU(2)$ instead of $SU(2) \times SU(2) \times SU(2)$, but not sure yet:
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 SuperconformalChern-Simons Theories (pdf)
Should also add pointer to
arxiv.org/abs/hep-th/0309106
and
arxiv.org/abs/hep-th/9906059
arxiv.org/abs/hep-th/9910100
added these pointers now:
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.
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