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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 (see below). The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, and a more complicated one, which yields a bialgebra, and can be extended with an antipode to a Hopf algebra structure. ''Note'': In this article, all algebras are assumed to be unital and associative. ==Construction== Let ''V'' be a vector space over a field ''K''. For any nonnegative integer ''k'', we define the ''k''th tensor power of ''V'' to be the tensor product of ''V'' with itself ''k'' times: : That is, ''T''''k''''V'' consists of all tensors on ''V'' of rank ''k''. By convention ''T''0''V'' is the ground field ''K'' (as a one-dimensional vector space over itself). We then construct ''T''(''V'') as the direct sum of ''T''''k''''V'' for ''k'' = 0,1,2,… : The multiplication in ''T''(''V'') is determined by the canonical isomorphism : given by the tensor product, which is then extended by linearity to all of ''T''(''V''). This multiplication rule implies that the tensor algebra ''T''(''V'') is naturally a graded algebra with ''T''''k''''V'' serving as the grade-''k'' subspace. This grading can be extended to a Z grading by appending subspaces for negative integers ''k''. The construction generalizes in straightforward manner to the tensor algebra of any module ''M'' over a ''commutative'' ring. If ''R'' is a non-commutative ring, one can still perform the construction for any ''R''-''R'' bimodule ''M''. (It does not work for ordinary ''R''-modules because the iterated tensor products cannot be formed.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「tensor algebra」の詳細全文を読む スポンサード リンク
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