15.10.2 Tensor Symmetric Algebra Product Relation

Tensor Symmetric Algebra Product Relation is the identification of the symmetric product multiplication on Sym(V) with the multiplication induced on the quotient of the full tensor algebra T(V) by the two-sided ideal generated by the differences u tensor v minus v tensor u, for all vectors u and v in V. This relation gives a second, algebraically intrinsic construction of the symmetric algebra, complementing the earlier construction via the symmetrization operator applied directly to tensor components, and it explains why the symmetric product is forced to be commutative: commutativity is not imposed after the fact but is built directly into the ideal being quotiented out.

Establishing this relation connects the symmetric tensor theory developed through permutation sums and symmetrization projectors to the more general algebraic technique of building a commutative algebra as a quotient of a non-commutative one, situating Sym(V) within the standard toolkit of quotient constructions used throughout abstract algebra.

Construction of the Ideal

Generators of the Ideal

Within the tensor algebra T(V), defined as the direct sum of all ordinary tensor powers of V, consider the two-sided ideal I generated by all elements of the form:

for every pair of vectors u and v in V; this ideal encodes precisely the failure of commutativity present in the raw, unsymmetrized tensor product.

The Ideal Consists of Elements Vanishing Under Symmetrization

Every generator u tensor v minus v tensor u lies in the kernel of the rank-two symmetrization operator, since Sym(u tensor v) and Sym(v tensor u) are both equal to u odot v by the commutative behavior of the symmetric product; because I is generated by these differences and Sym is linear, every element of I lies in the kernel of the full symmetrization operator across all ranks.

The Quotient Algebra

Defining the Quotient

The quotient algebra T(V) divided by I consists of equivalence classes of tensors, where two tensors are identified whenever their difference lies in I; the multiplication on this quotient is inherited directly from the tensor algebra's multiplication, namely the ordinary tensor product, applied to equivalence classes rather than to individual tensors.

Well-Definedness of the Inherited Multiplication

Because I is a two-sided ideal, multiplying two elements of T(V) that differ by something in I, by any third tensor, still produces a difference lying in I; this is exactly the algebraic condition required for the tensor product operation to descend to a well-defined multiplication on the quotient, independent of which representative is chosen from each equivalence class.

The Isomorphism to the Symmetrization Picture

Matching Quotient Classes to Symmetrized Tensors

The quotient map sending a tensor S in T(V) to its equivalence class in T(V) divided by I corresponds, under a natural identification, to sending S to its symmetrized image Sym(S); two tensors are identified in the quotient exactly when they have the same symmetrization, since their difference lying in I is equivalent to that difference lying in the kernel of Sym.

Multiplication Correspondence

Under this identification, the inherited quotient multiplication of two equivalence classes corresponds exactly to the symmetric product of the corresponding symmetrized tensors, since multiplying representatives in T(V) and then symmetrizing the result reproduces the same outcome as symmetrizing each representative first and then applying the symmetric product; this correspondence is precisely the content of the symmetric algebra product relation, confirming that the two constructions of Sym(V), one via direct symmetrization of components and one via quotienting the tensor algebra, agree completely.

Why the Ideal Forces Commutativity

Commutators as the Obstruction to Commutativity

The generators u tensor v minus v tensor u are exactly the commutators of the tensor algebra's multiplication restricted to elements of V; setting every such commutator to zero in the quotient is precisely what forces the resulting multiplication to be commutative, since any failure of commutativity in T(V) traces back, through repeated application of the product rule, to these elementary commutators between vectors.

Minimality of the Relation

The ideal I is generated only by commutators of vectors, not by commutators of higher-rank tensor products directly, yet the resulting quotient is fully commutative at every rank; this reflects the fact that commutators of higher-rank products can always be expressed, using the algebra's multiplication rules, in terms of commutators of the vector-level generators, so imposing commutativity at the vector level is sufficient to force it throughout the entire algebra.

Significance of the Quotient Construction

An Intrinsic, Basis-Free Definition

The quotient construction defines Sym(V) without reference to any chosen basis or to the explicit permutation-sum formula for symmetrization, relying only on the abstract tensor algebra and the notion of a two-sided ideal; this offers a construction of the symmetric algebra that is manifestly basis-independent from the outset, complementing the more computational, component-based approach used when symmetrization was first introduced.

Template for Related Constructions

The same quotient technique, replacing the commutator ideal with a different ideal generated by u tensor v plus v tensor u, produces the exterior algebra instead, underscoring that the symmetric algebra product relation is one instance of a general pattern in which imposing a chosen relation on the tensor algebra's generators determines the resulting quotient algebra's entire multiplicative structure.