Tensor algebra Definition and 68 Threads

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. cianfa72

    I Terminologies used to describe tensor product of vector spaces

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  2. jv07cs

    I Question Regarding Definition of Tensor Algebra

    I am currently reading this book on multilinear algebra ("Álgebra Linear e Multilinear" by Rodney Biezuner, I guess it only has a portuguese edition) and the book defines an Algebra as follows: It also defines the direct sum of two vector spaces, let's say V and W, as the cartesian product V x...
  3. What's a Tensor?

    What's a Tensor?

    A brief explanation of vector and tensor concepts from A Student's Guide to Vectors and Tensors by Dan Fleisch. I found this when I was trying to better understand tensors and how they are used.
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    • Media item
    • Tensor algebra Tensors
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    • Category: Linear Algebra
  4. G

    I Transfer rank2 tensor to a new basis

    The trace of the sigma should be the same in both new and old basis. But I get a different one. Really appreciate for the help. I’ll put the screen shot in the comment part
  5. T

    A How can I calculate the square of the Pauli-Lubanski pseudovector?

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  6. Vanilla Gorilla

    B Attempted proof of the Contracted Bianchi Identity

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  7. SH2372 General Relativity (8X): Components of tensor operations

    SH2372 General Relativity (8X): Components of tensor operations

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

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  9. A

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

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    Total Momentum Operator for Klein Gordon Field

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  11. L

    I Anyone knows why musical isomorphism is called so?

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  12. S

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  13. A

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  14. A

    I Deriving Contravariant Form of Levi-Civita Tensor

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  15. A

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  16. A

    I Purpose of Tensors, Indices in Tensor Calculus Explained

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  17. A

    I Expressing Vectors of Dual Basis w/Metric Tensor

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  18. Wannabe Physicist

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  19. S

    Quantum Tensor networks and tensor algebra

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  20. snypehype46

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  21. J

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  22. F

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  23. George Keeling

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  24. Data Base Erased

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  25. S

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  26. C

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  27. M

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  28. RicardoMP

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  29. T

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  30. George Keeling

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  31. Prez Cannady

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  32. T

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  33. Q

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  34. saadhusayn

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  35. Marcus95

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  36. Ron19932017

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  37. Avatrin

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  38. S

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  39. F

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  40. redtree

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  41. Math Amateur

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  42. F

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  43. Math Amateur

    I The Tensor Algebra - Cooperstein, Defn 10.5

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  44. F

    I Transformation of Tensor Components

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  45. P

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  46. V

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  47. S

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  48. D

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  49. C

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  50. K

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