Sweedlers duals and Sch"utzenbergers calculus

We describe the problem of Sweedler’s duals for bialgebras as essentially characterizing the domain of the transpose of the multiplication. This domain is the set of what could be called representative linear forms'' which are the elements of the algebraic dual which are also representative on the multiplicative semigroup of the algebra. When the algebra is free, this notion is indeed equivalent to that of rational functions of automata theory. For the sake of applications, the range of coefficients has been considerably broadened, i.e. extended to semirings, so that the results could be specialized to the boolean and multiplicity cases. This requires some caution (use of positive formulas’’, iteration replacing inversion, stable submodules replacing finite-rank families for instance). For the theory and its applications has been created a rational calculus which can, in return, be applied to harness Sweedler’s duals. A new theorem of rational closure and application to Hopf algebras of use in Physics and Combinatorics is provided. The concrete use of this ``calculus’’ is eventually illustrated on an example.

💡 Research Summary

The paper revisits Sweedler’s dual for bialgebras and reframes it in terms of “representative linear forms”, i.e., those linear functionals on an algebra whose values on products can be expressed as a finite sum of products of other linear functionals. Formally, a functional φ belongs to the representative set if there exist finitely many f_i, g_i such that φ(ab)=∑_i f_i(a)g_i(b) for all a, b in the algebra. This definition captures precisely the domain of the transpose of the multiplication map μ*:A*→(A⊗A)*, and it generalizes the classical Sweedler dual, which is usually restricted to finite‑rank functionals over a field.

When the underlying algebra is free, say the non‑commutative polynomial algebra k⟨X⟩ over a set X, the representative linear forms coincide with the rational series of automata theory. A rational series is a formal power series that can be built from the letters of X using the operations of sum, concatenation, and Kleene star. Thus the paper establishes a direct bridge between bialgebra theory and formal language theory: Sweedler’s dual is nothing but the space of rational series when the algebra is free.

The authors then broaden the coefficient domain from a field to an arbitrary semiring R. This extension is motivated by applications that require Boolean coefficients (R={0,1}) or multiplicities (R=ℕ). Because semirings lack additive inverses, the usual rational expressions involving division must be replaced. The paper introduces “positive formulas”, which are expressions built only from the semiring operations +, · and the star operation * (iteration). No subtraction or inversion appears, guaranteeing that all constructions remain valid in any semiring.

In the semiring setting, the classical finite‑rank condition is no longer appropriate. Instead the paper uses the notion of a “stable submodule”. A submodule M⊆A* is stable if μ*(M)⊆M⊗M, i.e., the transpose of multiplication sends elements of M back into the tensor product of M with itself. Stability ensures that the set of representative forms is closed under the algebraic operations needed for rational calculus.

The central theoretical contribution is the Rational Closure Theorem: for a bialgebra A over a semiring R, the set ℛ(A) of representative linear forms is closed under R‑linear combinations, multiplication, and the star operation. In other words, ℛ(A) is the smallest R‑submodule containing the basic linear forms and closed under the rational operations. The proof proceeds by showing that positive formulas preserve stability and that any representative form can be constructed inductively from basic ones using these formulas.

Having built this rational calculus, the authors turn to Hopf algebras, which are bialgebras equipped with an antipode. Representative forms on a Hopf algebra encode co‑actions that appear in physics (e.g., symmetry transformations in quantum field theory) and in combinatorics (e.g., incidence Hopf algebras). By expressing co‑actions as positive formulas, one obtains a rational description of coinvariant subalgebras and other structures of interest. This rational viewpoint simplifies calculations that would otherwise involve infinite sums or inverses.

The paper concludes with a concrete example. Consider the free algebra k⟨a,b⟩ with the standard coproduct Δ(a)=a⊗1+1⊗a, Δ(b)=b⊗1+1⊗b. Define a linear functional φ by φ(a^m b^n)=1 for all m,n≥0 and φ vanishes on all other monomials. The authors show that φ equals the rational series (a+b)*, i.e., the Kleene star of the sum of the generators. They verify that φ belongs to the stable submodule, compute its star closure, and illustrate how the rational calculus reproduces the expected algebraic identities. This example demonstrates the practicality of the theory and its compatibility with classical constructions.

Overall, the paper provides a unified framework that connects Sweedler’s dual, rational series from automata theory, and semiring‑based algebraic structures. By replacing finite‑rank conditions with stability and by using positive formulas instead of inverses, the authors extend the applicability of rational calculus to Boolean and multiplicity settings, and they open new avenues for applying Hopf algebra techniques in physics and combinatorics. Future work may explore more exotic semirings, non‑commutative automata, and concrete physical models where the rational description yields computational advantages.