2.5 Enveloping algebras

Enveloping algebras bridge Lie theory and associative algebra, extending Lie algebras while preserving their properties. They're crucial for studying representations and provide a more manageable framework for complex Lie algebra concepts.

The universal enveloping algebra U(g) of a Lie algebra g allows free multiplication of elements without brackets. The Birkhoff-Witt theorem gives U(g) a concrete basis, helping us understand its structure and dimension.

Definition of enveloping algebras

• Enveloping algebras extend Lie algebras to associative algebras preserving crucial properties
• Fundamental concept in Non-associative Algebra bridging Lie theory and associative algebra
• Provides a framework to study representations of Lie algebras in a more tractable setting

Universal enveloping algebras

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• Associative algebra U(g) constructed from a Lie algebra g
• Preserves the Lie bracket structure of g within U(g)
• Contains g as a Lie subalgebra via the canonical embedding
• Allows multiplication of Lie algebra elements freely without brackets

Birkhoff-Witt theorem

• Establishes a vector space basis for universal enveloping algebras
• States that U(g) has a basis consisting of ordered monomials in a basis of g
• Provides a concrete realization of the abstract universal enveloping algebra
• Crucial for understanding the structure and dimension of U(g)

Construction of enveloping algebras

• Two primary methods used to construct enveloping algebras from Lie algebras
• Both approaches yield isomorphic results but offer different insights
• Understanding these constructions deepens comprehension of enveloping algebras in Non-associative Algebra

Tensor algebra approach

• Utilizes the tensor algebra T(g) of the underlying vector space of g
• Forms U(g) as a quotient of T(g) by the ideal generated by xy - yx - [x,y]
• Preserves the universal property of the tensor algebra
• Allows for a natural gradation on the enveloping algebra

Quotient algebra method

• Starts with the free algebra on the basis elements of g
• Imposes relations corresponding to the Lie bracket structure of g
• Results in an algebra satisfying the universal property of enveloping algebras
• Provides a more direct link to the original Lie algebra structure

Properties of enveloping algebras

• Enveloping algebras possess rich structural properties derived from their construction
• These properties make them powerful tools in Non-associative Algebra and representation theory
• Understanding these properties is crucial for applying enveloping algebras effectively

Universal property

• Any Lie algebra homomorphism from g to L(A) extends uniquely to an algebra homomorphism U(g) to A
• Characterizes U(g) up to isomorphism
• Facilitates the study of Lie algebra representations through associative algebra representations
• Provides a bridge between Lie theory and associative algebra theory

PBW basis

• Poincaré-Birkhoff-Witt basis provides an explicit vector space basis for U(g)
• Consists of ordered monomials in a basis of g
• Allows for explicit computations and dimension calculations in U(g)
• Crucial for understanding the structure of U(g) as a vector space

Filtration and grading

• Natural filtration on U(g) induced by the degree of monomials in the PBW basis
• Associated graded algebra Gr(U(g)) isomorphic to the symmetric algebra S(g)
• Provides a connection between U(g) and polynomial algebras
• Useful for studying the structure and representations of U(g)

Applications in representation theory

• Enveloping algebras play a central role in the representation theory of Lie algebras
• Provide a bridge between Lie algebra representations and associative algebra modules
• Essential tools for studying infinite-dimensional representations in Non-associative Algebra

Lie algebra representations

• Every U(g)-module corresponds to a g-module and vice versa
• Allows the use of associative algebra techniques in Lie algebra representation theory
• Facilitates the construction and analysis of irreducible representations
• Provides a framework for studying weight modules and highest weight theory

Module structures

• U(g)-modules offer a richer structure than g-modules
• Allow for the definition of primitive ideals and the study of their properties
• Provide a setting for analyzing the category of g-modules using homological algebra
• Enable the development of character theory for infinite-dimensional representations

Enveloping algebras vs Lie algebras

• Comparison highlights the advantages of working with enveloping algebras
• Illustrates the interplay between associative and non-associative structures in algebra
• Crucial for understanding the role of enveloping algebras in Non-associative Algebra

Structural differences

• U(g) is associative while g is non-associative
• U(g) has infinite dimension (for non-zero g) while g may be finite-dimensional
• U(g) contains a copy of g as a Lie subalgebra
• Multiplication in U(g) allows for "commuting" elements of g without using brackets

Algebraic advantages

• U(g) allows for the use of well-developed associative algebra techniques
• Provides a setting for applying homological algebra methods to Lie algebra problems
• Facilitates the study of infinite-dimensional representations
• Allows for the definition and study of concepts like primitive ideals

Hopf algebra structure

• Enveloping algebras naturally possess a Hopf algebra structure
• This structure connects enveloping algebras to quantum groups and deformation theory
• Provides additional tools for studying representations and duality in Non-associative Algebra

Comultiplication and counit

• Comultiplication Δ: U(g) → U(g) ⊗ U(g) defined by Δ(x) = x ⊗ 1 + 1 ⊗ x for x ∈ g
• Counit ε: U(g) → k defined by ε(1) = 1 and ε(x) = 0 for x ∈ g
• These operations make U(g) a bialgebra
• Allow for the definition of tensor products of representations

Antipode map

• Antipode S: U(g) → U(g) defined by S(x) = -x for x ∈ g
• Satisfies the antipode axioms making U(g) a Hopf algebra
• Provides a notion of "inverse" elements in U(g)
• Crucial for defining dual representations and studying duality in representation theory

Examples of enveloping algebras

• Concrete examples illustrate the general theory of enveloping algebras
• Provide insight into the structure and representations of important Lie algebras
• Essential for developing intuition about enveloping algebras in Non-associative Algebra

U(sl2)

• Enveloping algebra of the special linear Lie algebra sl2
• Generated by elements E, F, H with relations [H,E] = 2E, [H,F] = -2F, [E,F] = H
• Plays a crucial role in the representation theory of sl2
• Serves as a building block for understanding more complex enveloping algebras

U(gl(n))

• Enveloping algebra of the general linear Lie algebra gl(n)
• Generated by elements Eij with relations [Eij, Ekl] = δjkEil - δilEkj
• Important in the study of matrix Lie algebras and their representations
• Provides a framework for understanding the representation theory of gl(n) and related Lie algebras

Generalizations and variants

• Extensions of the enveloping algebra concept to other algebraic structures
• Illustrate the broader applicability of enveloping algebra ideas in Non-associative Algebra
• Provide connections to quantum theory and supersymmetry

Quantum enveloping algebras

• Deformations of classical enveloping algebras depending on a parameter q
• Arise in the theory of quantum groups and quantum integrable systems
• Preserve many properties of classical enveloping algebras in a deformed setting
• Important in mathematical physics and representation theory

Super enveloping algebras

• Enveloping algebras of Lie superalgebras
• Incorporate Z2-grading to accommodate both even and odd elements
• Crucial in the study of supersymmetry and superalgebra representations
• Extend enveloping algebra techniques to the context of graded algebras

Computational aspects

• Practical considerations for working with enveloping algebras
• Essential for applying enveloping algebra theory to concrete problems
• Highlight the interplay between theoretical and computational aspects of Non-associative Algebra

Gröbner bases

• Provide a systematic approach to computations in enveloping algebras
• Allow for the solution of the ideal membership problem and other algebraic tasks
• Crucial for implementing computer algebra systems for enveloping algebras
• Enable effective calculations with PBW bases and filtrations

Computer algebra systems

• Software packages (GAP, Magma) implement enveloping algebra computations
• Facilitate exploration of examples and testing of conjectures
• Allow for the study of high-dimensional enveloping algebras
• Provide tools for calculating with representations and module structures