Download Communications in Mathematical Physics - Volume 244 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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H J is “dexter” antisymmetric} (31) using Eq. (8). We now explain what we mean by a dexter matrix. Say G is a 2N × 2N orthogonal antisymmetric matrix. Then an orthogonal change of basis puts it into the canonical form JN . However, this may not be possible by means of a proper orthogonal change of basis (i. , of determinant +1). Specifically, when N is even, the two complex structures ±JN are equivalent (under, say, the proper orthogonal change of basis JN as in (21)), but when N is odd they are not.

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R ), |λj | = 1. (41) Abusing notation, we will also write R for the vector (λ1 , . . , λR ). The tangent space a to this torus at the identity is identified with the space of R-tuples i = (iθ1 , . . , iθR ), θj ∈ R. Recall that we identify p with the tangent space to G/K at the base-point o = K/K. The exponential maps Exp of G/K and exp of G are related by Exp(X) = exp(X)K ∈ G/K (42) for X ∈ p = To (G/K). For i ∈ a, exp(i ) is given by the matrix on the right-hand side of Eqs. (27), (32), (35) and (40), respectively, provided we choose λj = eiθj in (41).

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