Lie algebras contained in Universal Enveloping Algebra

In summary, the question is about identifying Lie algebras in the universal enveloping algebra of a particular Lie algebra. The specific example given is a Lie algebra of the form [A,B] = ηB, and the goal is to find all isomorphic copies of this Lie algebra in the algebra generated by A and B with the relation AB - BA = ηB. Possible resources for finding an answer include PBW or its corollaries.
  • #1
Couchyam
122
18
I've got a general question about Lie algebras, which is basically this:
Q: What is there to be said about the Lie algebras that can be identified in the universal enveloping algebra of a particular Lie algebra?
E.g. if I have a Lie algebra of the form
[A,B] = ηB,
then I would like to identify all of the Lie algebras that can be found in the algebra generated by A and B with the relation
AB - BA = ηB.

I would be very happy if someone could direct me to any resources related to this question. I am sorry that the question is somewhat vague, but if something resembling an answer forms in the back of your mind please don't hesitate to post.
 
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  • #2
If I understand this correctly, then you look for all isomorphic copies of ##\mathfrak{g}## in ##\mathfrak{U(g)}##. I'm not sure, but maybe PBW or one of its corollaries could help.
 

1. What is a Lie algebra?

A Lie algebra is a mathematical structure that studies the properties of vector spaces and their related operations, such as addition and multiplication. It is used to analyze the algebraic properties of continuous symmetry groups in physics and other areas of mathematics.

2. What is a Universal Enveloping Algebra (UEA)?

A Universal Enveloping Algebra is the algebraic structure that contains all the information about a given Lie algebra. It is a way of embedding a Lie algebra into an associative algebra, allowing for the use of more powerful algebraic tools and techniques.

3. How are Lie algebras contained in Universal Enveloping Algebras?

Lie algebras are contained in Universal Enveloping Algebras through a process known as the universal enveloping construction. This involves taking the original Lie algebra and defining new operations on it in order to embed it into the larger algebraic structure of the UEA.

4. What are the applications of Lie algebras contained in Universal Enveloping Algebras?

Lie algebras contained in Universal Enveloping Algebras have many applications in mathematics and physics. They are used to study symmetry groups, differential equations, and other algebraic structures. They also have applications in theoretical physics, such as in the study of quantum mechanics and string theory.

5. How are Lie algebras and Universal Enveloping Algebras related to other algebraic structures?

Lie algebras and Universal Enveloping Algebras are closely related to other algebraic structures, such as associative algebras, commutative algebras, and Lie groups. They also have connections to representation theory, which studies how abstract algebraic structures can be represented by matrices and other linear operators.

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