Tensor algebra

The tensor algebra is a mathematical term used in many areas of mathematics such as linear algebra , algebra , differential geometry, and physics . It combines "all tensors " over a vector space in the structure of a graduated algebra .

definition

Let it be a vector space over a body or, more generally, a module over a commutative ring with one element. Then the tensor algebra (as a vector space) is defined by the direct sum of all tensor products of the space with itself. ${\displaystyle V}$ ${\displaystyle K}$

${\displaystyle \mathrm {T} (V):=\bigoplus _{n\geq 0}V^{\otimes n}=K\oplus V\oplus (V\otimes V)\oplus (V\otimes V\otimes V)\oplus \ldots }$

With the multiplication, which is given by the tensor product on the homogeneous components , becomes a - graduated , unitary, associative algebra . ${\displaystyle \mathrm {T} (V)}$ ${\displaystyle \mathbb {N} _{0}}$

Universal property

The tensor satisfies the following universal property: Is an associative - algebra with a unit element , and a linear map, so there is exactly one Algebrenhomomorphismus so that the chart ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle e}$ ${\displaystyle f\colon V\to A}$ ${\displaystyle {\tilde {f}}\colon \mathrm {T} (V)\to A}$

commutes. This algebra homomorphism is given by as well . ${\displaystyle {\tilde {f}}(v_{1}\otimes \dots \otimes v_{r})=f(v_{1})\dots f(v_{r})}$ ${\displaystyle {\tilde {f}}(\lambda )=\lambda e}$

T as functor

${\displaystyle \mathrm {T} }$is a functor from the category of -vector spaces into the category of -algebras. For a vector space homomorphism (a linear mapping) is given by the algebra homomorphism, which is induced by (here is the embedding) according to the universal property of the tensor algebra . ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle \varphi \colon V\to W}$ ${\displaystyle \mathrm {T} (\varphi ):\mathrm {T} (V)\to \mathrm {T} (W)}$ ${\displaystyle i_{W}\circ \varphi :V\to \mathrm {T} (W)}$ ${\displaystyle i_{W}:W\to \mathrm {T} (W)}$

The functor is left adjoint to the forgetting functor , which assigns the underlying vector space to an -algebra . Hence it is also referred to as the free algebra about . ${\displaystyle \mathrm {T} }$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle \mathrm {T} (V)}$ ${\displaystyle V}$

example

If a -dimensional vector space (or a free module of rank ) is then isomorphic to the free associative algebra over in indefinite. In the case is isomorphic to the polynomial ring . ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle \mathrm {T} (V)}$ ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle n=1}$ ${\displaystyle \mathrm {T} (V)}$ ${\displaystyle K[X]}$

If, more generally, is an arbitrary non-empty set and if the vector space generated via , that is, the free K -module is above , then the freely generated associate algebra is. ${\displaystyle X}$ ${\displaystyle V_{X}}$ ${\displaystyle X}$ ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle \mathrm {T} (V_{X})}$ ${\displaystyle X}$

Quotient spaces of tensor algebra

By dividing out a certain ideal , one can derive from the tensor algebra , for example, the symmetrical algebra , the outer algebra or the Clifford algebra . These algebras are important in differential geometry .