By V. A. Krechmar

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**Example text**

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 .

### A Problem Book in Algebra by V. A. Krechmar

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