Tensor algebra

A part of tensor calculus in which algebraic operations on tensors (cf. Tensor on a vector space) are studied.

The tensor algebra of a unitary module $V$ over a commutative associative ring $A$ with unit is the algebra $T(V)$ over $A$ whose underlying module has the form

$$ T(V) = \bigoplus_{p=0}^\infty T^{p, 0}(V) = \bigoplus_{p=0}^\infty \bigotimes^p V $$

and in which multiplication is defined with the help of tensor multiplication (cf. Tensor on a vector space). Besides the contravariant tensor algebra, the covariant tensor algebra

$$ T(V^*) = \bigoplus_{p=0}^\infty T^{0, p}(V) $$

is also considered, as well as the mixed tensor algebra

$$ \widehat T(V) = \bigoplus_{p, q = 0}^\infty T^{p, q}(V) . $$

If the module $V$ is free and finitely generated, then $T(V^*)$ is naturally isomorphic to the algebra of all multilinear forms (cf. Multilinear form) on $V$. Any homomorphism $V \to W$ of $A$-modules naturally defines a tensor algebra homomorphism $T(V) \to T(W)$.

The tensor algebra $T(V)$ is associative, but in general not commutative. Its unit is the unit of the ring $A = T^0(V)$. Any $A$-linear mapping of the module $V$ into an associative $A$-algebra $B$ with a unit can be naturally extended to a homomorphism of algebras $T(V) \to B$ mapping the unit to the unit. If $V$ is a free module with basis $(v_i)_{i \in I}$, then $T(V)$ is the free associative algebra with system of generators $(v_i)_{i \in I}$.

References

[1]	N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[2]	A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian)