The tensor algebra of a vector space 
over a field 
is the direct sum
 |
where 
and 
is the 
-fold
vector space 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.
More generally, for a unital R-module 
over a commutative unit ring 
, the tensor algebra is the graded
ring
 |
where 
and 
is the 
-fold
module tensor product over 
. Multiplication concatenates
factors. Taking 
allows the construction for any Abelian group (The
Stacks Project Authors 2026).
