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There is a theorem of Rosenlicht ("Some basic theorems on algebraic groups", 1956, Theorem 13) asserting that a quotient of a connected algebraic group by its center is linear. So a connected algebraic group with trivial center is linear.

Is it true of connected complex Lie groups? I.e. are is a connected complex Lie groups group with a trivial center linear a subgroup of $GL(n,\mathbb{C})$? Is it algebraic?

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There is a theorem of Rosenlicht ("Some basic theorems on algebraic groups", 1956, Theorem 13) asserting that a quotient of a connected algebraic group by its center is linear. So a connected algebraic group with trivial center is linear.

Is it true of connected complex Lie groups? I.e. are connected algebraic complex Lie groups with trivial center linear algebraic?

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# is connected complex Lie group with a trivial center linear?

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