A semisimple group is a type of algebraic group that has no nontrivial connected solvable normal subgroups, making it a central object of study in the theory of Lie groups and algebraic groups. It is characterized by its rich structure and the fact that its representation theory is well-behaved, allowing for a comprehensive classification of its finite-dimensional representations.

Semisimple Group

An algebraic group is a group that is also an algebraic variety, meaning it has a compatible structure of both a group and a geometric object defined by polynomial equations. This dual nature allows algebraic groups to be studied using techniques from both algebra and geometry, making them fundamental objects in areas such as number theory, representation theory, and algebraic geometry.

Algebraic Group

Nontrivial connected solvable normal subgroups

Theory of Lie groups

Representation Theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. This approach allows complex algebraic problems to be translated into more manageable linear algebra problems, providing deep insights into the symmetry and structure of mathematical objects.

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