nLab
enveloping algebra

There is also a distinct notion of an enveloping algebra of a Lie algebra.


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Definition

Given a monoid AA with multiplication μ\mu in a symmetric monoidal category CC with symmetry τ\tau its opposite A opA^{op} is the same underlying object AA with multiplication μτ\mu\circ\tau. The enveloping monoid A eA^e of AA is the monoid whose underlying object is AA opA\otimes A^{op} and for which the multiplication is given by

(μ(μτ))(idτid):(AA op)(AA op)AA op (\mu\otimes (\mu\circ\tau))\circ(id\otimes\tau\otimes id): (A\otimes A^{op})\otimes (A\otimes A^{op})\to A\otimes A^{op}

where τ=τ A,A\tau=\tau_{A,A} is the symmetry AAAAA\otimes A\to A\otimes A.

The enveloping monoid is sometimes called the enveloping algebra, especially if the monoidal category is a category of vector spaces. The left A eA^e-modules in CC are in 1-1 correspondence with the AA-AA-bimodules in CC.

Revised on January 24, 2013 17:51:00 by Urs Schreiber (82.113.99.233)