Why is Z not a semisimple?

The most basic example of a semisimple module is a module over a field, i.e., a vector space. On the other hand, the ring Z of integers is not a semisimple module over itself, since the submodule 2Z is not a direct summand. Let A be an algebra over a field K.

How do you prove that Lie algebra is semisimple?

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras (non-abelian Lie algebras without any non-zero proper ideals).

Who invented Lie algebra?

Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the 1870s, and independently discovered by Wilhelm Killing in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group is used.

What is Lie algebra used for?

Abstract Lie algebras are algebraic structures used in the study of Lie groups. They are vector space endomorphisms of linear transformations that have a new operation that is neither commutative nor associative, but referred to as the bracket operation, or commutator.

Is every representation completely reducible?

4. Any finite dimensional representation of A is completely reducible (that is, isomorphic to a direct sum of irreducible representations).

Is the trivial representation irreducible?

The trivial representation of a finite group G is C equipped with the trivial action of G: gx = x for every x in C and for every g in G. Note that every finite group has the trivial representation, and since C has no proper nontrivial subspaces, it is irreducible, as is any one-dimensional representation.

Is Z an Injective module?

The factor group Q/Z and the circle group are also injective Z-modules. The factor group Z/nZ for n > 1 is injective as a Z/nZ-module, but not injective as an abelian group.

What are the Semisimple Z-modules?

By Chinese remainder theorem, semisimple Z-modules are precisely the Z/nZ where n is a square-free integer. If F is a field, then Fn is certainly semisimple as an F-module. In fact any semisimple F-module is isomorphic to Fn for some n i.e. it is a finite dimensional vector space.

What is Lie theory?

In mathematics, the mathematician Sophus Lie (/ˈliː/ LEE) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. The subject is part of differential geometry since Lie groups are differentiable manifolds.

How do you say Lie group?

In mathematics, a Lie group (pronounced /liː/ “Lee”) is a group that is also a differentiable manifold.

Is a circle a Lie group?

Informally, a Lie group is a group of symmetries where the symmetries are continuous. A circle has a continuous group of symmetries: you can rotate the circle an arbitrarily small amount and it looks the same. This is in contrast to the hexagon, for example.

Is Lie algebra a group?

Definition 2.1. A Lie group is an algebraic group (G, ⋆) that is also a smooth manifold, such that: (1) the inverse map g ↦→ g−1 is a smooth map G → G.