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Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → U < sub > L </ sub >, ( notation as above ) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → A < sub > L </ sub > there exists a unique unital algebra homomorphism g: U → A such that: f (-) = g < sub > L </ sub > ( h (-)).

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## Some Related Sentences

Let and X

"

__Let____X__**be****the**unit Cartesian square ×,**and**let ~**be****the**equivalence relation on__X__defined by ∀**a**, b ∈ ((**a**, 0**)**~**(****a**, 1**)**∧**(**0, b**)**~**(**1, b )).__Let__

**(**

__X__

**<**

**sub**

**>**i

**</**

**sub**

**>,**

**f**

**<**

**sub**

**>**ij

**</**

**sub**>)

**be**an inverse system

**of**objects

**and**morphisms in

**a**category C

**(**same definition

**as**

**above**).

*

__Let__**the**index set I**of**an inverse system**(**__X__**<****sub****>**i**</****sub****>,****f****<****sub****>**ij**</****sub**>) have**a**greatest element m**.**Then**the**natural projection π**<****sub****>**m**</****sub**>:__X__**→**__X__**<****sub****>**m**</****sub****>****is**an isomorphism**.**

Let and be

__Let__every policeman

**and**park guard keep his eye on John

**and**Jane Doe, lest one piece

**of**bread

__be__placed undetected

**and**one bird survive

**.**

__Let__us assume

**that**

**it**would

__be__possible

**for**an enemy to create an aerosol

**of**

**the**causative agent

**of**epidemic typhus

**(**Rickettsia prowazwki

**)**

**over**City

**A**

**and**

**that**

**a**large number

**of**cases

**of**typhus fever resulted therefrom

**.**

__Let__p

__be__

**the**minimal polynomial

**for**T, Af, where

**the**Af, are distinct irreducible monic polynomials

**over**F

**and**

**the**Af are positive integers

**.**

__Let__V

__be__

**a**finite-dimensional vector space

**over**an algebraically closed field F, e.g.,

**the**field

**of**complex numbers

**.**

__Let__N

__be__

**a**positive integer

**and**let V

__be__

**the**space

**of**all N times continuously differentiable functions F on

**the**real line which satisfy

**the**differential equation Af where Af are some fixed constants

**.**

__Let__Q

__be__

**a**nonsingular quadric surface bearing reguli Af

**and**Af,

**and**let **zg

__be__

**a**Af curve

**of**order

**K**on Q

**.**

__Let__us take

**a**set

**of**circumstances in which I happen to

__be__interested on

**the**legislative side

**and**in which I think every one

**of**us might naturally make

**such**

**a**statement

**.**

__Let__

**the**state

**of**

**the**stream leaving stage R

__be__denoted by

**a**vector Af

**and**

**the**operating variables

**of**stage R by Af

**.**

__Let__

**it**

__be__granted then

**that**

**the**theological differences in this area between Protestants

**and**Roman Catholics appear to

__be__irreconcilable

**.**

__Let__us therefore put first things first,

**and**make sure

**of**preserving

**the**human race at whatever

**the**temporary price may

__be__''

**.**

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