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 (see below).
The tensor algebra is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure.
Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.
the tensoralgebra 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...
two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense...
manipulations involving the tensor and wedge symbols. This distinction is developed in greater detail in the article on tensoralgebras. Here, there is much...
mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map...
mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring...
algebra concepts find applications in various areas, including: Classical treatment of tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket...
of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensoralgebra of a module, allowing one to define multiplication...
diagonalization. The free algebra generated by V may be written as the tensoralgebra ⨁n≥0 V ⊗ ⋯ ⊗ V, that is, the direct sum of the tensor product of n copies...
category of R-algebras. Tensor products The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras for more...
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components...
symmetric algebra. As the symmetric algebra of a vector space is a quotient of the tensoralgebra, an element of the symmetric algebra is not a tensor, and...
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T ( v σ 1 , v...
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold...
and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used...
multilinear algebra. Ricci calculus The earliest foundation of tensor theory – tensor index notation. Order of a tensor The components of a tensor with respect...
tensor is antisymmetric with respect to its first three indices. If a tensor changes sign under exchange of each pair of its indices, then the tensor...
enveloping algebras. The construction proceeds by first building the tensoralgebra of the underlying vector space of the Lie algebra. The tensoralgebra is simply...
essentially the composition of functions. In the language of tensoralgebra, a particular tensor is associated with a particular shape with many lines projecting...
tensor algebra T ( g ) {\displaystyle T({\mathfrak {g}})} from it. The tensoralgebra is a free algebra: it simply contains all possible tensor products...
In multilinear algebra, a tensor decomposition is any scheme for expressing a "data tensor" (M-way array) as a sequence of elementary operations acting...
multilinear algebra, the tensor rank decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is minimal...
referred to as their tensor product, and can be used to define the tensoralgebra. The outer product contrasts with: The dot product (a special case of...
at p. When the exterior algebra is viewed as a quotient of the tensoralgebra, the exterior product corresponds to the tensor product (modulo the equivalence...
metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite if g(v, v) > 0 for...
Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products...
relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of...
category of R-algebras to the category of sets. Free algebras over division rings are free ideal rings. Cofree coalgebra Tensoralgebra Free object Noncommutative...
property is used rather than the concrete details. For example, the tensoralgebra of a vector space is slightly complicated to construct, but much easier...
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