By Dragan Milicic

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Algebra. Rings, modules and categories

VI of Oregon lectures in 1962, Bass gave simplified proofs of a couple of "Morita Theorems", incorporating principles of Chase and Schanuel. one of many Morita theorems characterizes whilst there's an equivalence of different types mod-A R::! mod-B for 2 jewelry A and B. Morita's answer organizes principles so successfully that the classical Wedderburn-Artin theorem is a straightforward final result, and additionally, a similarity type [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring ok comprises all algebras B such that the corresponding different types mod-A and mod-B together with k-linear morphisms are similar through a k-linear functor.

Matrix Partial Orders, Shorted Operators and Applications (Series in Algebra)

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Geometry and Algebra in Ancient Civilizations

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Extra info for Lectures on Lie Groups

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Hence, it has finitely many connected components. Clearly, for a matrix T ∈ O(n), T ∗ is its transpose. , det(T ) = ±1. It follows that the homomorphism det maps O(n) onto the subgroup {±1} of R∗ . Therefore, SO(n) is a normal Lie subgroup of O(n) of index 2. In particular, SO(n) is open in O(n). Moreover, it has finitely many connected components. The natural action of the group O(n) on Rn preserves the euclidean distance. Therefore, it acts differentiably on the unit sphere S n−1 = {x ∈ Rn | x21 + x22 + · · · + x2n = 1}.

A Lie algebra a is abelian if [x, y] = 0 for all x, y ∈ a. Let a be a Lie algebra. We denote by aopp the opposite Lie algebra of a. It is the same linear space with the commutator (x, y) −→ [x, y]◦ = −[x, y]. Clearly, aopp is a Lie algebra. Moreover, x −→ −x is an isomorphism of a with aopp . Evidently, we have (aopp )opp = a. If dim a = 1, a has to be abelian. If dim a = 2, we can pick a basis (v1 , v2 ) of a and see that [x, y] is proportional to [v1 , v2 ] for any x, y ∈ a. Therefore, we can assume that [x, y] is proportional to e1 for any x, y ∈ a.

7. Lemma. Let ϕ : G −→ H be a Lie group homomorphism of a simply ˜ be the connected, connected Lie group G into a connected Lie group H. Let H ˜ universal covering Lie group of H and p : H −→ H the covering projection. Then ˜ such that p ◦ ϕ˜ = ϕ. there exists a unique Lie group homomorphism ϕ˜ : G −→ H In addition, if ϕ : G −→ H is a Lie group morphism of connected Lie groups, ˜ −→ H ˜ such that the diagram there exists a unique Lie group homomorphism ϕ˜ : G ϕ ˜ ˜ −−− G −→   pG ϕ ˜ H  pH , G −−−−→ H where the vertical arrows are covering projections, is commutative.