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Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.
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Groups of matrices that preserve several quadratic forms
Let $V$ be the vector space you begin with. As you probably know, transformations $T \in \operatorname{End}(V)$ that preserve a symmetric form $(-,-)$ of full rank are called orthogonal, and the grou …