Linear representations in Char 0

In summary, in characteristic zero, any linear representation of a reductive group or a finite group is semisimple, according to Maschke's Theorem. However, it is uncertain if any linear representation of any group is semisimple in characteristic zero. To explore this possibility, we can consider the integers under addition, which is not finite and has a single generator. This allows for the assignment of a matrix to represent a 2-dimensional representation, and further investigation is needed to determine if it is irreducible.
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In characteristic zero any linear representation of a reductive group is semisimple. Also in characteristic zero any linear representation of a finite group is semisimple (Maschke's Thm). However is any linear representation of any group semisimple in characteristic zero?
 
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Evidently we must try to think of a group that is not finite, nor an algebraic group (let's avoid all of them). What is there to think of? First we hit the integers under addition - it is not finite, and not defined by a finite number of equations as a subset of R/C/Q/ any field, so we've got a hope. It is generated by a single element so that is good - a rep is just assigning it to a matrix and we're done. From there you should try to think about what it means for a 2-d rep, say (hint!) to be irreducible (again, hint, do all matrices have two eigenvectors?).
 
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No, not all linear representations of groups are semisimple in characteristic zero. While it is true that any linear representation of a reductive group or a finite group is semisimple in characteristic zero, this does not apply to all groups. There are examples of groups, such as the Heisenberg group, where not all linear representations are semisimple in characteristic zero. This is because the property of being semisimple is dependent on the structure of the group itself, not just the characteristic of the field in which it is being represented. So while characteristic zero does have some nice properties for linear representations, it does not guarantee semisimplicity for all groups.
 

What is a linear representation?

A linear representation is a way of describing a mathematical object, such as a group or vector space, using linear transformations. This means that the object can be represented by matrices and operations such as addition and multiplication can be performed on these matrices.

What is Char 0 in linear representations?

Char 0 refers to the characteristic of a field, which is the smallest number of times that the field's multiplicative identity must be added to itself to equal zero. In linear representations, Char 0 means that the field has characteristic zero, and thus does not have any elements that can be added to themselves a finite number of times to equal zero.

Why is Char 0 important in linear representations?

Char 0 is important in linear representations because it allows for the use of infinite dimensional vector spaces, which are necessary for certain types of mathematical objects. It also simplifies calculations and proofs, as there are no elements that can be added to themselves a finite number of times to equal zero.

What are some examples of linear representations in Char 0?

Some examples of linear representations in Char 0 include representations of finite groups, representations of Lie algebras, and representations of topological groups. These all involve using linear transformations to describe the structures and operations of these mathematical objects.

What are the applications of linear representations in Char 0?

Linear representations in Char 0 have many applications in mathematics, physics, and engineering. They are used to study and understand the properties of mathematical objects and their applications in various fields, such as quantum mechanics, differential equations, and coding theory.

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