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. The tensor algebra is, in a sense, the "most general" algebra containing V. This notion of generality is formally expressed by a certain universal property (see below). Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

Note: In this article, all algebras are assumed to be unital and associative. In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ...

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: In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

That is, T^kV consists of all tensors on V of rank k. By convention T⁰V is the ground field K (as a one-dimensional vector space over itself). Note: This is a fairly abstract mathematical approach to tensors. ...

We then construct T(V) as the direct sum of T^kV for k = 0,1,2,… In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

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^kV serving as the grade-k subspace. In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ...

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.) In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. ...

Universal property

The fact that the tensor algebra is the most general algebra containing V is expressed by the following universal property: Any linear transformation f : V → A from V to an algebra A over K can be uniquely extended to an algebra homomorphism from T(V) to A as indicated by the following commutative diagram: In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... A homomorphism between two algebras over a field K, A and B, is a map such that for all k in K and x,y in A, F(kx)=kF(x) F(x+y)=F(x)+F(y) F(xy)=F(x)F(y) Categories: Math stubs | Algebra ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

Here i is the canonical inclusion of V into T(V). One can, in fact, define the tensor algebra T(V) as the unique algebra satisfying this property (specifically, it is unique up to a unique isomorphism). Image File history File links Universal property of the tensor algebra of a vector space V. TeX source begin{diagram} V & rTo^{i;} & T(V) & rdTo_{f} & dDashTo>{;tilde f} & & A end{diagram} File links The following pages link to this file: Tensor algebra ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ...

The above universal property shows that the construction of the tensor algebra is functorial in nature. That is, T is a functor from the K-Vect, category of vector spaces over K, to K-Alg, the category of K-algebras. The functoriality of T means that any linear map from V to W extends uniquely to an algebra homomorphism from T(V) to T(W). Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, the category K_Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms. ...

The tensor algebra T(V) is also called the free algebra on the vector space V. As with other free constructions, the functor T is left adjoint to some forgetful functor, specifically the functor which sends each K-algebra to its underlying vector space. In abstract algebra, a free algebra is the noncommutative analogue of a polynomial ring. ... The idea of a free object in mathematics is one of the basics of abstract algebra. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... A forgetful functor is a type of functor in mathematics. ...

If V has finite dimension n, another way of looking at the tensor algebra is as the "algebra of polynomials over K in n non-commuting variables". If we take basis vectors for V, those become non-commuting variables (or indeterminants) in T(V), subject to no constraints (beyond associativity, the distributive law and K-linearity). In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

Quotients

Because of the generality of the tensor algebra, many other algebras of interest are constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain quotients of T(V). Examples of this are the exterior algebra, the symmetric algebra, Clifford algebras and universal enveloping algebras. In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... Clifford algebras are a type of associative algebra in mathematics. ... In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ...