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No.

For instance, topologically $U(2) = SU(2) \times U(1)$, since both are homeomorphic to $S^3 \times S^1$, but the group structures are different. Another example is given by $SO(3) \times SU(2)$ which is diffeomorphic to $SO(4)$.

On the other hand, any commutative compact real Lie group of dimension $n$ is isomorphic (as a real Lie group) to the real torus $\mathbb{T}^n:=(\mathbb{S}^1)^n$.

Analogously, any commutative compact complex Lie group of dimension $n$ is isomorphic (as a complex Lie group) to a complex torus, i.e. a quotient of the form $\mathbb{C}^n/\Gamma$, where $\Gamma \subset \mathbb{C}^n$ is a lattice. Two complex tori $\mathbb{C}^n /\Gamma_1$ and $\mathbb{C}^n / \Gamma_2$ are isomorphic as complex Lie groups if and only if there exists $g \in \textrm{GL}_n (\mathbb{C})$ such that $\Gamma_2=g (\Gamma_1)$, but of course they are always both isomorphic to $\mathbb{T}^{2n}$ as real Lie groups.

No.

For instance, topologically $U(2) = SU(2) \times U(1)$, since both are homeomorphic to $S^3 \times S^1$, but the group structures are different. Another example is given by $SO(3) \times SU(2)$ which is diffeomorphic to $SO(4)$.

On the other hand, any commutative connected compact real Lie group of dimension $n$ is isomorphic (as a real Lie group) to the real torus $\mathbb{T}^n:=(\mathbb{S}^1)^n$.

Analogously, any connected compact complex Lie group of dimension $n$ is isomorphic (as a complex Lie group) to a complex torus, i.e. a quotient of the form $\mathbb{C}^n/\Gamma$, where $\Gamma \subset \mathbb{C}^n$ is a lattice. Notice that in the compact complex case commutativity comes for free. Two complex tori $\mathbb{C}^n /\Gamma_1$ and $\mathbb{C}^n / \Gamma_2$ are isomorphic as complex Lie groups if and only if there exists $g \in \textrm{GL}_n (\mathbb{C})$ such that $\Gamma_2=g (\Gamma_1)$, but of course they are always both isomorphic to $\mathbb{T}^{2n}$ as real Lie groups.

No.

For instance, topologically $U(2) = SU(2) \times U(1)$, since both are homeomorphic to $S^3 \times S^1$, but the group structures are different. Another example is given by $SO(3) \times SU(2)$ which is diffeomorphic to $SO(4)$.

No.

For instance, topologically $U(2) = SU(2) \times U(1)$, since both are homeomorphic to $S^3 \times S^1$, but the group structures are different. Another example is given by $SO(3) \times SU(2)$ which is diffeomorphic to $SO(4)$.

On the other hand, any commutative compact real Lie group of dimension $n$ is isomorphic (as a real Lie group) to the real torus $\mathbb{T}^n:=(\mathbb{S}^1)^n$.

Analogously, any commutative compact complex Lie group of dimension $n$ is isomorphic (as a complex Lie group) to a complex torus, i.e. a quotient of the form $\mathbb{C}^n/\Gamma$, where $\Gamma \subset \mathbb{C}^n$ is a lattice. Two complex tori $\mathbb{C}^n /\Gamma_1$ and $\mathbb{C}^n / \Gamma_2$ are isomorphic as complex Lie groups if and only if there exists $g \in \textrm{GL}_n (\mathbb{C})$ such that $\Gamma_2=g (\Gamma_1)$, but of course they are always both isomorphic to $\mathbb{T}^{2n}$ as real Lie groups.

No.

For instance, topologically $U(2) = SU(2) \times U(1)$ but the group structures are different. Another example is given by $SO(3) \times SU(2)$ which is diffeomorphic to $SO(4)$.

No.

For instance, topologically $U(2) = SU(2) \times U(1)$, since both are homeomorphic to $S^3 \times S^1$, but the group structures are different. Another example is given by $SO(3) \times SU(2)$ which is diffeomorphic to $SO(4)$.