What is a Lie algebra, really

A geometric view
The standard view: a linear space L with a product, i.e., a bilinear map L x L ---> L denoted 

a,b ---->  [a,b]

Let us however view it as a linear space L with a (1,2)-variant tensor c. We may view c as a map 


L* x L x L ----> R
Different restrictions of this map to a fewer number of arguments result in major concepts of Lie agebra: 

Algebra product       m:    L/\L ---> L                  [v,w] = c( . ,v , w)

Co-algebra product   m*:      L* ---> L* /\ L*       m*(alpha) = c(alpha, . , . )

Adjoint map          ad:      L  ---> L x L*             ad(v) = c( . , v, . )

Tong map            tau:  L* x L ---> L*          tau(alpha,v) = c (alpha, v, . )

These maps can be put into one diagram:

graph of a Lie map
This is the mandala of a Lie algebra.

Click here
to get to an interactive page, where you can click on a particular item
of the mandala to get further definitions.


Copyright © 1995 jerzy kocik

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Lie alg
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