Tensor algebra - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector 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). In Mathematics, especially in the area of Abstract algebra known as Ring theory, a free algebra is the noncommutative analogue of a Polynomial ring In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism

The tensor algebra also has a coalgebra structure.

Note: In this article, all algebras are assumed to be unital and associative. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) 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

1 Construction
2 Adjunction and universal property
3 Non-commutative polynomials
4 Quotients
5 Coalgebra structure
6 See also

Construction

Let V be a vector space over a field K. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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:

$T^kV = V^{\otimes k} = V\otimes V \otimes \cdots \otimes V.$

That is, T^kV consists of all tensors on V of rank k. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually By convention T⁰V is the ground field K (as a one-dimensional vector space over itself).

We then construct T(V) as the direct sum of T^kV for k = 0,1,2,…

$T(V)= \bigoplus_{k=0}^\infty T^kV = K\oplus V \oplus (V\otimes V) \oplus (V\otimes V\otimes V) \oplus \cdots.$

The multiplication in T(V) is determined by the canonical isomorphism

$T^kV \otimes T^\ell V \to T^{k + \ell}V$

given by the tensor product, which is then extended by linearity to all of T(V). The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction 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 (or Commutative ring) with an extra piece of structure

The construction generalizes in straightforward manner to the tensor algebra of any module M over a commutative ring. 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 In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property If R is a non-commutative ring, one can still perform the construction for any R-R bimodule M. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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 (It does not work for ordinary R-modules because the iterated tensor products cannot be formed. )

Adjunction and universal property

The tensor algebra T(V) is also called the free algebra on the vector space V, and is functorial. In Mathematics, especially in the area of Abstract algebra known as Ring theory, a free algebra is the noncommutative analogue of a Polynomial ring As with other free constructions, the functor T is left adjoint to some forgetful functor, here the functor which sends each K-algebra to its underlying vector space. In Mathematics, the idea of a free object is one of the basic concepts of Abstract algebra. In Mathematics, in the area of Category theory, a forgetful functor is a type of Functor.

Explicitly, the tensor algebra satisfies the following universal property, which formally expresses the statement that it is the most general algebra containing V:

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:

Universal property of the tensor algebra

Here i is the canonical inclusion of V into T(V) (the unit of the adjunction). In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that A homomorphism between two algebras over a field K, A and B, is a map FA\rightarrow B such that for all k In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also 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), but one must still prove that an object satisfying this property exists. 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. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, especially Category theory, the category K-Vect has all Vector spaces over a fixed field K as objects 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).

Non-commutative polynomials

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). Basis vector redirects here For basis vector in the context of crystals see Crystal structure. 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

Note that the algebra of polynomials on V is not $T (V)$ , but rather $T (V *)$ : a (homogeneous) linear function on V is an element of $V *$ .

Quotients

Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i. e. by constructing certain quotient algebras of T(V). In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the Examples of this are the exterior algebra, the symmetric algebra, Clifford algebras and universal enveloping algebras. In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field In Mathematics, Clifford algebras are a type of Associative algebra. In Mathematics, for any Lie algebra L one can construct its universal enveloping algebra U ( L)

Coalgebra structure

The coalgebra structure on the tensor algebra is given as follows. In Mathematics, coalgebras are structures that are dual to Unital Associative algebras The Axioms of unital associative algebras can The coproduct Δ is defined by

$\Delta(v_1 \otimes \dots \otimes v_m ) := \sum_{i=0}^{m} (v_1 \otimes \dots \otimes v_i) \otimes (v_{i+1} \otimes \dots \otimes v_m)$

extended by linearity to all of TV. The counit is given by ε(v) = 0-graded component of v. Note that Δ : TV → TV ⊗ TV respects the grading

$T^mV \to \bigoplus_{i+j=m} T^iV \otimes T^jV$

and ε is also compatible with the grading.

The tensor algebra is not a bialgebra with this coproduct. In Mathematics, a bialgebra over a field K is a structure which is both a Unital Associative algebra and a Coalgebra over However, the following more complicated coproduct does yield a bialgebra:

$\Delta(x_1\otimes\dots\otimes x_m) = \sum_{p=0}^m \sum_{\sigma\in\mathrm{Sh}_{p,m-p}} \left(v_{\sigma(1)}\otimes\dots\otimes v_{\sigma(p)}\right)\otimes\left(v_{\sigma(p+1)}\otimes\dots\otimes v_{\sigma(m)}\right)$

where the summation is taken over all (p,m-p)-shuffles. Finally, the tensor algebra becomes a Hopf algebra with antipode given by

$S(x_1\otimes\dots\otimes x_m) = (-1)^mx_m\otimes\dots\otimes x_1$

extended linearly to all of TV. In Mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( Unital associative algebra, a Coalgebra

Contents

Construction

Adjunction and universal property

Non-commutative polynomials

Quotients

Coalgebra structure

See also