Lie group homomorphisms and the adjoint representation

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Problem II.7.1.2 from Кострикин А.И. "Линейная алгебра" [Кос00]More problems in Section II.7.1 "The norm and functions of a linear operator"

Construct an isomorphism of the groups $U(1)$ and $S O(2, \mathbb {R})$ .

Problem III.3.6.4 from Кострикин А.И. "Основные структуры алгебры" [Кос04]More problems in Section III.3.6 "Representations of the groups $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$"

Show, without appealing to the complete description of the irreducible representations of the group $\mathrm{SO}(3)$ , that a homomorphism $\tau : \mathrm{SO}(3) \longrightarrow \mathrm{SU}(2)$ can only be trivial.

Problem 12.2.1 from Винберг Э.Б. "Курс алгебры" [Вин14]More problems in Section 12.2 "The Exponential Map"

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Problem 12.2.2 from Винберг Э.Б. "Курс алгебры" [Вин14]More problems in Section 12.2 "The Exponential Map"

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Problem 12.3.1 from Винберг Э.Б. "Курс алгебры" [Вин14]More problems in Section 12.3 "The Tangent Lie Algebra and the Adjoint Representation"

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Problem 12.3.2 from Винберг Э.Б. "Курс алгебры" [Вин14]More problems in Section 12.3 "The Tangent Lie Algebra and the Adjoint Representation"

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Problem 12.3.5 from Винберг Э.Б. "Курс алгебры" [Вин14]More problems in Section 12.3 "The Tangent Lie Algebra and the Adjoint Representation"

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Problem 12.4.2 from Винберг Э.Б. "Курс алгебры" [Вин14]More problems in Section 12.4 "Linear Representations of Lie Groups"

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Problem 12.4.3 from Винберг Э.Б. "Курс алгебры" [Вин14]More problems in Section 12.4 "Linear Representations of Lie Groups"

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Problem 12.4.4 from Винберг Э.Б. "Курс алгебры" [Вин14]More problems in Section 12.4 "Linear Representations of Lie Groups"

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References

[Кос00] Кострикин А.И. Линейная алгебра. ФИЗМАТЛИТ, 2000.

[Кос04] Кострикин А.И. Основные структуры алгебры. ФИЗМАТЛИТ, 2004.

[Вин14] Винберг Э.Б. Курс алгебры. 2-е изд., МЦНМО, 2014.