Tensor algebra Definition and 46 Discussions

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.

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  1. Vanilla Gorilla

    B Attempted proof of the Contracted Bianchi Identity

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    • Orodruin
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    • Tensor algebra
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    • Category: Relativity
  3. SH2372 General Relativity (8): Tensor operations

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    • Orodruin
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    • Tensor algebra Tensor analysis
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  4. A

    Showing that the gradient of a scalar field is a covariant vector

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  5. Samama Fahim

    Total Momentum Operator for Klein Gordon Field

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  6. L

    I Anyone knows why musical isomorphism is called so?

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  7. S

    I Proving ##\partial^{i} = g^{ik} \partial_{k}##

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  8. A

    I Parallel transport of a tensor

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  9. A

    I Derivation of the contravariant form for the Levi-Civita tensor

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  10. A

    I Showing that the determinant of the metric tensor is a tensor density

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  11. A

    I Doubt about the purpose of some elements in tensor calculus

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  12. A

    I Expressing the vectors of the dual basis with the metric tensor

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  13. Wannabe Physicist

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  14. snypehype46

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  15. J

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  16. F

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  17. George Keeling

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  18. Data Base Erased

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  19. S

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  20. C

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  21. M

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  23. T

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  25. Prez Cannady

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  26. T

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  27. Q

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  28. saadhusayn

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  29. Marcus95

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  30. Ron19932017

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  31. Avatrin

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    • Avatrin
    • Thread
    • definition tensor algebra
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    • Forum: Calculus
  32. F

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  33. redtree

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  34. F

    I What is the outer product of a tensor product of vectors?

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  35. F

    I Transformation of Tensor Components

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  36. P

    I Mathematics of tensor products in the Bell states

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  37. V

    I Understanding dual basis

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  38. S

    Tensor product of two arbitrary vectors an arbitrary tensor?

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  39. D

    Deriving geodesic equation using variational principle

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  40. C

    Divergence of Gradient inverse

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  41. K

    Transformation rule for product of 3rd, 2nd order tensors

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  42. meyol99

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  43. D

    Definition of tensor product

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  44. BiGyElLoWhAt

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  45. sweetdreams12

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  46. C

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