Tensoralgebra

The tensor algebra is a mathematical term that is used in many areas of mathematics such as linear algebra, algebra, differential geometry and in physics. It summarizes " all tensors " on a vector space in the structure of a graded algebra together.

Definition

It is a vector space over a field or more generally a module over a commutative ring with unit element. Then the tensor algebra (as a quantity ) is defined by the direct sum of all tensor products of the room with himself.

With the multiplication is given to the homogeneous components of the tensor product, is a -graded, unitary associative algebra.

Universal property

Is an associative - algebra with a unit element, and a linear map, then there exists exactly one algebra homomorphism such that the diagram

Commutes. This algebra homomorphism is given by and.

This universal property shows that a functor from the category of K- vector spaces is in the category of K- algebras. Forms the functor

On

From.

Example

Is one -dimensional vector space (or a free module of rank ), it is isomorphic to the free associative algebra over in indeterminates.

Quotient spaces of the tensor algebra

By removing parts of a certain ideal of the tensor algebra one can, for example, the symmetric algebra, the exterior algebra or Clifford algebra win. These algebras are of importance in differential geometry.

  • Algebra (Structure )
  • Algebra
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