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In mathematics, the tensor algebra of a vector space V, denoted T(V) or T • (V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product.It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property ...
Tensor algebra In the tensor algebra T(V) of a vector space V, the operation becomes a normal (internal) binary operation. A consequence is that T(V) has infinite dimension unless V has dimension 0. The free algebra on a set X is for practical purposes the same as the tensor algebra on the vector space with X as basis.
For example, the tensor algebra construction on a vector space is the left adjoint to the functor on associative algebras that ignores the algebra structure. It is therefore often also called a free algebra. Likewise the symmetric algebra and exterior algebra are free symmetric and anti-symmetric algebras on a vector space.
A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.
In mathematics, a monoidal category (or tensor category) is a category equipped with a bifunctor ⊗ : C × C → C {\displaystyle \otimes :\mathbf {C} \times \mathbf {C} \to \mathbf {C} } that is associative up to a natural isomorphism , and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism.
Xerus [52] is a C++ tensor algebra library for tensors of arbitrary dimensions and tensor decomposition into general tensor networks (focusing on matrix product states). It offers Einstein notation like syntax and optimizes the contraction order of any network of tensors at runtime so that dimensions need not be fixed at compile-time.
The map L → U(L) of K-modules canonically extends to a map T(L) → U(L) of algebras, where T(L) is the tensor algebra on L (for example, by the universal property of tensor algebras), and this is a filtered map equipping T(L) with the filtration putting L in degree one (actually, T(L) is graded).
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor [1]) is a tensor that can be written as a product of tensors of the form = where a, b, ..., d are nonzero and in V or V ∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors.