- #1
HDB1
- 77
- 7
Please, I have a question about this:
The Universal enveloping algebra of a finite dimensional Lie algebra is Noetherian.How we can prove it? Please..
HDB1 said:Please, I have a question about this:
The Universal enveloping algebra of a finite dimensional Lie algebra is Noetherian.
How we can prove it? Please..
It is a ##\mathbb{K}##-vector space and as such a ##\mathbb{K}##-module. We say vector space and finite-dimensional in case the scalars are from a field, and we say module and finitely generated in case the scalars are from a ring, e.g. the integers.HDB1 said:Thank you so much, @fresh_42 , please, why Universal enveloping algebra is module? PBW theorem gives a basis of Universal enveloping algebra, but please, why it is finite dimensional? please,
I thougt in general: lie lagebra is finite dimensioal, and its universal enveloping is infinite dimensional.
Thanks in advance,
A universal enveloping algebra is a mathematical structure that is used to study Lie algebras, which are algebraic structures that describe the symmetry of a system. The universal enveloping algebra is constructed by taking the tensor algebra of the Lie algebra and then quotienting out by a specific ideal. This allows for the study of the Lie algebra in a more manageable and structured way.
The universal enveloping algebra is useful in the study of Lie algebras because it provides a way to understand the structure and properties of the Lie algebra. It also allows for the calculation of invariants and other important quantities associated with the Lie algebra. Additionally, the universal enveloping algebra has applications in physics, particularly in the study of symmetries in quantum mechanics.
Yes, any finite-dimensional Lie algebra over a field of characteristic 0 can have a universal enveloping algebra. However, for infinite-dimensional Lie algebras, the existence of a universal enveloping algebra is not guaranteed and depends on certain conditions being satisfied.
A universal enveloping algebra is different from a group algebra in that it is specifically used to study Lie algebras, while a group algebra is used to study group representations. Additionally, a universal enveloping algebra is a non-commutative algebra, while a group algebra is commutative.
Universal enveloping algebras have applications in a variety of fields, including physics, representation theory, and algebraic geometry. They are also used in the study of differential equations and non-commutative geometry. In physics, universal enveloping algebras are used to study symmetries in quantum mechanics and to understand the behavior of particles. In representation theory, they are used to classify representations of Lie algebras. In algebraic geometry, they are used to study the geometry of algebraic varieties.