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A

Symplectic groups and

The quaternions are the H = 1,i ,i ,i = R4 with h 1 2 3i ∼ i1i2 = i2i1 = i3,i2i3 = i3i2 = i1,i3i1 = i1i3 = i2 and i2 = i2 = i2 = 1. If− we have a − x = x + 3− x i then its 1 2 3 − 0 j=1 j j conjugate is given byx ¯ = x 3 x i . The symplectic , Sp(n) is 0 j=1 j j P the of Gl(n, H), the− invertible n n quaternionic matrices, which preserves the standard HermitianP form on×Hn: n x, y = x¯ y . h i i i i=1 X This means that if A Sp(n), A¯T A = AA¯T = 11, it is the quaternionic U(n, H),∈ sometimes called the hyperunitary group. Sp(n) is a real of dimension n(2n + 1), compact and (simply) connected. The is given by the n n quaternionic matrices B that satisfy × B + B¯T =0 . A different, but closely related, type of is Sp(2n, F ) the group of degree 2n over a field F , in other words, the group of 2n 2n symplectic matrices with entries in F and with group operation that× of multiplication. If F is the field of real/complex numbers the Lie group Sp(2n, F ) has real/complex dimension n(2n + 1). Since all symplectic matrices have unit , the symplectic group is a subgroup of the special Sl(2n, F ), in fact for n = 1 this means that Sp(2, F )= Sl(2, F ). If D Sp(2n, F ) then ∈ 0 11 DT D = with = . (A.1) C C C 11 0  −  The Lie algebra of Sp(2n, F ) is given by the set of 2n 2n matrices E (over F ) that satisfy × E + ET =0 . C C

111 112 A Symplectic groups and quaternions

The symplectic group Sp(2n, F ) can also be defined as the set of linear transformations of a 2n-dimensional (over F ) that preserve a nondegenerate antisymmetric . This precisely leads to the property (A.1). Unitary-symplectic groups are the intersection of unitary groups and sym- plectic groups, both acting in the same vector space. That is, the elements of the unitary-symplectic groups are elements in both unitary and sym- plectic groups: USp(n, n)= U(n, n; C) Sp(2n; C) . ∩ Explicitly we can construct such groups by considering (A.1) with D a 2n 2n complex matrix which has to satisfy the additional condition × 11n n 0 D† D = with = × . (A.2) H H H 0 11n n  − ×  The general block form of this D is T V ? D = (A.3) V T ?   making (A.2) equivalent to

T †T V †V = 11 ?− ? T †V V †T =0 . − These groups are isomorphic to symplectic groups in the hyperunitary sense:

USp(2n) ∼= Sp(n) . Finally there is the following R Sp(2n; ) ∼= USp(2n) , (A.4) R from which we see that Sp(2n; ) ∼= Sp(n). A quaternionic is Riemannian but not necessarily complex, which means that the group Sp(n) Sp(1) is contained in O(4n) as a subgroup. On the other hand, hyperk¨ahler× are in fact K¨ahler (with respect to all three complex structures, see also (1.5)) and must therefore have their holonomy contained in U(2n). In terms of holonomy groups, the difference between the two types of manifolds is the factor of Sp(1). This arises from the fact that (1.12) is defined up to local SO(3) ∼= Sp(1) rotations. This feature is absent in the hyperk¨ahler case since the complex structures are covariantly constant and globally defined. For more information on the relation between holonomy groups, symplectic groups and quaternions, see [23, 39, 68, 35, 132].