The tensor algebra of a vector space 
over a field 
is the direct sum
 |
where 
and 
is the 
-fold
tensor product of 
. Multiplication is concatenation
of tensor factors, making 
an associative algebra.
Every linear transformation from 
to an associative algebra
extends uniquely to an algebra homomorphism from
. A universal
enveloping algebra is obtained by quotienting a tensor algebra by relations determined
by a Lie bracket.
