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Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
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Do two dimensional representations with the same adjoint representation differ by a character?
Let $K$ be a field of characteristic not equal to $2$. Let $\text{ad} : \text{GL}_2(K) \to \text{GL}_3(K)$ be the adjoint representation, obtained by $\text{GL}_2(K)$ acting on $2 \times 2$ matrices …