Universal enveloping algebra u(g)

Definition

The universal enveloping algebra u(g) is an associative algebra constructed from a given Lie algebra g, serving as a bridge between the realm of Lie algebras and representation theory. It allows for the extension of representations of Lie algebras to representations of associative algebras, enabling the use of powerful algebraic tools to study their structure and representations. This construction is essential in understanding how Lie algebras can be represented in a more manageable framework.

5 Must Know Facts For Your Next Test

1. The universal enveloping algebra u(g) is defined as the quotient of the tensor algebra T(g) by the two-sided ideal generated by elements of the form x ensor y - y ensor x - [x,y], where x and y are elements of g.
2. u(g) retains a natural grading based on the degree of elements, which allows for easier handling of both representations and computations involving the algebra.
3. For any representation of a Lie algebra g, there exists an induced representation on u(g), providing a powerful method to study representations using tools from associative algebra.
4. The center of u(g), consisting of elements that commute with every element in u(g), plays a crucial role in classifying irreducible representations of the Lie algebra g.
5. The universal enveloping algebra is finite-dimensional over its center when g is a finite-dimensional semisimple Lie algebra, which connects deeply with important results like the Borel-Weil theorem.

Review Questions

• How does the construction of the universal enveloping algebra u(g) from a Lie algebra g facilitate the study of representations?
  • The universal enveloping algebra u(g) provides an associative structure that extends the representations defined on the Lie algebra g. By allowing elements of g to act on vector spaces through linear transformations within u(g), it simplifies many complex problems associated with studying these representations. This connection enables mathematicians to apply techniques and results from associative algebra to analyze and classify representations of Lie algebras more effectively.
• Discuss the significance of the center of u(g) in relation to irreducible representations of g.
  • The center of u(g) is crucial because it consists of elements that commute with all other elements in u(g). When studying irreducible representations of a Lie algebra g, elements in the center act as scalars on these representations. This means that understanding the structure of the center helps classify irreducible representations and reveals information about how these representations behave under various actions.
• Evaluate how the universal enveloping algebra u(g) contributes to connecting various branches of mathematics such as representation theory and non-associative algebra.
  • The universal enveloping algebra u(g) serves as a pivotal link between Lie algebras and representation theory, showcasing how abstract concepts can be applied in concrete settings. By translating problems in non-associative structures into associative settings through u(g), it opens avenues for using powerful techniques from commutative and non-commutative algebra. This interconnection not only enhances our understanding of Lie algebras but also impacts areas such as quantum mechanics and mathematical physics where these structures frequently arise.

Related terms

Lie Algebra: A vector space equipped with a binary operation called the Lie bracket that satisfies bilinearity, antisymmetry, and the Jacobi identity.

Representation: A homomorphism from a Lie algebra to the algebra of linear transformations on a vector space, providing a way to study the algebra's action on that space.

Jacobi Identity:

An important property of Lie algebras stating that for any elements x, y, z in the algebra, the relation [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 holds.