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Universal enveloping algebra

Definition

Let $g$ be a Lie algebra over $k$ . In $T (g)$ , regard $[x, y] \in g$ as an element of tensor degree one, and let $I$ be the two-sided ideal generated by

$x \otimes y - y \otimes x - [x, y] (x, y \in g) .$

The universal enveloping algebra is

$U (g) = T (g) / I .$

Write

$ι_{g} : g ⟶ U (g)$

for the composite of the degree-one inclusion with the quotient map. This definition makes $U (g)$ a unital associative algebra and $ι_{g}$ a linear map. It does not assert that $ι_{g}$ is injective; that conclusion will come from PBW.

Depends on

Used by

Arakawa-Suzuki Casimir operators on tensor functors Definition
Casimir operator relative to an invariant form Definition
Dominant cyclic highest-weight presentation Definition
PBW filtration on the enveloping algebra Definition
The deformed tensor space for the type-A finite W-algebra Definition
The shifted Yangian and the pyramid generators of the type-A finite W-algebra Definition
The type-A finite W-algebra and the Whittaker category Definition
The type-A nilpotent pyramid, good grading and Whittaker character Definition
The Whittaker functor from the relevant parabolic category-O summand Definition
The Casimir element in U(sl₂) Example
Verma modules for sl2 Example
Injectivity is not part of the enveloping quotient definition False statement
Verma modules need not be finite-dimensional False statement
An augmented coideal ideal of an enveloping algebra is generated by its primitive part Lemma
Gan-Ginzburg and Skryabin inputs for the type-A pyramid good grading Lemma
The canonical map to U(g) is a Lie homomorphism Lemma
The dominant cyclic generator survives Lemma
The opposite Borels of a symmetrizable Kac–Moody algebra are root-degreewise dual Lie bialgebras Lemma
The parabolic highest-weight truncation factors through the cyclotomic quotient Lemma
Unique simple quotient of the dominant cyclic module Lemma
Chevalley–Eilenberg cohomology computes Ext of the trivial module Proposition
The enveloping algebra of an abelian Lie algebra is symmetric Proposition
Ado's theorem with nilpotent nilradical action Theorem
Lie representations are U(g)-modules Theorem
Ordered monomial PBW basis for the shifted Yangian Theorem
Polynomial Weyl algebra presentation and normal-ordered basis Theorem
Skryabin's equivalence for the type-A Whittaker pair Theorem
The formal quantum Serre half embeds in the shuffle algebra and is degreewise free Theorem
Universal property of the enveloping algebra Theorem
Vust's graded double centralizer for the type-A nilpotent centralizer Theorem

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Etingof, MIT 18.745 notes, §12.1, printed pp. 69–70 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §5.1, printed pp. 71–72 (standard reference, not scraped)