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Recall that for matrices, the exponential map is defined by ∞ exp(x) = 0 xk . k! 6) It is well-known that this power series converges and defines an analytic map gl(n, K) → gl(n, K), where gl(n, K) is the set of all n × n matrices. In a similar way, we define the logarithmic map by ∞ log(1 + x) = 1 (−1)k+1 xk . 7) So defined, log is an analytic map defined in a neighborhood of 1 ∈ gl(n, K). The following theorem summarizes the properties of exponential and logarithmic maps. Most of the properties are the same as for numbers; however, there are also some differences due to the fact that multiplication of matrices is not commutative.

P as a shorthand for (Rp−1 )∗ v). p. Let us use this to construct H . 48. For every point g ∈ G, there is locally an integral manifold of the distribution Dh containing g, namely H 0 · g, where H 0 = exp u for some neighborhood u of 0 in h. This lemma can be easily proved using Frobenius theorem. Indeed, the distribution Dh is generated by right-invariant vector fields corresponding to elements of h. 28), this shows that the space of fields tangent to Dh is closed under the commutator, and thus Dh is completely integrable.

4. 25). Show that Fn (C) = GL(n, C)/B(n, C) = U(n)/T (n) where B(n, C) is the group of invertible complex upper triangular matrices, and T (n) is the group of diagonal unitary matrices (which is easily shown to be the n-dimensional torus (R/Z)n ). Deduce from this that Fn (C) is a compact complex manifold and find its dimension over C. 5. Let Gn,k be the set of all dimension k subspaces in Rn (usually called the Grassmanian). Show that Gn,k is a homogeneous space for the group O(n, R) and thus can be identified with coset space O(n, R)/H for appropriate H .

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A Problem Book in Algebra by V. A. Krechmar

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