Veronese Powers of Operads

• Veronese powers of operads are defined as suboperads generated by fixed-weight operations, extending the classical Veronese subalgebra construction.
• They yield (m+1)-ary analogues of binary structures but generally do not improve homological properties as seen in associative algebras.
• Quadratic Veronese powers establish a link with Koszul duality, providing a foundation for pure homotopy algebras and explicit operadic presentations.

A Veronese power of an operad is a suboperad generated by operations of a specific fixed weight, generalizing the Veronese subalgebra construction for algebras to the context of operads. Formally, for a connected, nonnegatively weight-graded operad 𝒫 over a field of characteristic zero, and integer m ≥ 1, the m-th Veronese power 𝒫^{[m]} is the suboperad generated by 𝒫^{(m)}, the component of all operations of total weight m. This construction provides an operadic framework for defining (m+1)-ary analogues of structures originally presented by binary operads, with significant roles in the theory of strongly homotopy algebras and Koszul duality. While the classical Veronese construction for associative algebras typically improves homological properties, the analogous operadic construction exhibits a markedly different behavior, failing in general to enhance such properties. Notable applications include the construction of Lie k-algebras, Lie triple systems, and their Koszul dual commutative multi-ary algebras, and the explicit presentation of ternary dendriform systems.

1. Foundational Definitions and Construction

Let 𝒫 = {𝒫(n)}{n≥0} be a weight-graded (possibly nonsymmetric) operad. The decomposition 𝒫(n) = ⊕{k≥0} 𝒫(n)^{(k)} defines weight spaces, and operadic composition preserves total weight. The m-th Veronese power 𝒫^{[m]} := ⟨𝒫^{(m)}⟩ is the smallest suboperad of 𝒫 containing all weight-m operations. In particular, for a binary operad 𝒱, such as the associative or dendriform operad, 𝒱^{[m]} is generated by composites of exactly m copies of the binary generators, yielding a family of canonical (m+1)-ary operations. The "naïve" Veronese power Va(𝒫) = ⊕_{k≥0} 𝒫^{(km)} with component-wise operadic structure, does not coincide with 𝒫^{[m]} in general, as it may not be generated by 𝒫^{(m)}. For example, in the free operad on a commutative binary operation, Va(𝒯) is not generated by its weight-m component for m≥2, as partial compositions are insufficient to generate all required monomials (Dotsenko et al., 2017).

2. Homological Properties and Their Divergence from the Associative Case

In associative, weight-graded algebras, Veronese subalgebras A^{[d]} typically enhance homological behavior: minimal relations become lower-degree, Koszulness may eventually be achieved for large d, and quadratic Gröbner bases appear. In the operadic context, these phenomena are absent in general. Explicit constructions demonstrate that, for weight-graded operads 𝒪:

• The minimal relation weights in 𝒪^{[d]} need not decrease;
• The off-diagonal bar homology does not necessarily improve;
• High Veronese powers may lack quadratic Gröbner bases, even when 𝒪 itself possesses one.

A central observation is that the process of composing weight-d generating operations fails, in the operadic setting, to recapitulate the full combinatorial structure needed to approximate the classical Veronese improvements. This discrepancy is exemplified by the free operad on one commutative binary operation, where certain tree monomials of weight 2d cannot be expressed as composites of elements of weight d (Dotsenko et al., 2017).

3. Koszul Duality and Pure Homotopy-Algebra Operads

Restoring a meaningful duality involves passing to the quadratic Veronese suboperad: q 𝒫^{[m]} = T(𝒫^{(m)}) / ⟨ quadratic relations among 𝒫^{(m)} ⟩ where T(𝒫^{(m)}) is the free operad on 𝒫^{(m)}. The central result asserts an isomorphism of dg-operads: (q 𝒫^{[m]})^! ≅ (𝒫^!)_{(m)} where 𝒫^! is the Koszul dual cooperad and (𝒫^!)_{(m)} its subcooperad cogenerated by weight-m elements [(Dotsenko et al., 2017), Theorem 32]. This establishes that quadratic Veronese powers of Koszul operads are Koszul dual to "pure" homotopy-algebra operads in which only one operation of a given arity is nontrivial.

For a connected weight-graded cooperad ℚ, the weight-k Koszul dual operad ℚ^!_{(k)} is the cobar construction Ωℚ with generators restricted to weight k. Algebras governed by these operads are termed pure strongly homotopy 𝒫-algebras, each with a unique nonzero structure map of a fixed weight.

Selected examples:

• For the associative operad Ass, (q Ass^{[k]})^! is the operad of partially associative (k+1)-ary algebras.
• For the commutative operad Com, (q Com^{[k]})^! is the operad governing L_infty-algebras with a single nontrivial bracket, corresponding to Lie_{(k+1)} (Dotsenko et al., 2017).

4. Classical Examples: Lie Triple Systems and Lie k-Algebras

Lie triple systems and Lie k-algebras are salient examples illustrating the utility of Veronese powers in encoding higher-arity algebraic structures:

• The operad Lie^{[2]}, regarded under the arity-minus-one weight grading, is the suboperad generated by Lie(3); this coincides with the operad LTS of Lie triple systems. The operations satisfy full skew-symmetry, the ternary Jacobi identity, and a derivation law:
  • [a_2,a_1,a_3] = -[a_1,a_2,a_3],
  • [a_1,a_2,a_3] + [a_2,a_3,a_1] + [a_3,a_1,a_2] = 0,
  • [a_1,a_2,[a_3,a_4,a_5]] = [[a_1,a_2,a_3],a_4,a_5] + [a_3,[a_1,a_2,a_4],a_5] + [a_3,a_4,[a_1,a_2,a_5]].
  • This operad is quadratic and Koszul, with Koszul dual being Com^{(3)}, the totally commutative ternary operad (Dotsenko et al., 2017).
• More generally, for each k ≥ 2, the operad Lie^{[k]}=⟨ Lie^{(k)} ⟩ yields the "Lie k-algebra" operad, quadratic and Koszul, whose Koszul dual is Com^{(k+1)}, governing totally commutative (k+1)-ary algebras.

5. Veronese Squares of Binary Operads: The Dendriform Case

For a binary operad 𝒱, 𝒱^{[2]} (the Veronese square) organizes the structure of ternary analogues:

• In the dendriform operad 𝒟, generated by binary operations x⋖y, x⋗y subject to three relations, dim 𝒟(3)=5; five non-leading monomials are selected as ternary generators ω_1,...,ω_5.
• 𝒟^{[2]}, the Veronese square, is then the suboperad generated by ω_1,...,ω_5.
• The embedding of 𝒟 into the Rota–Baxter operad RB via ε(x⋖y) = x U(y), ε(x⋗y) = U(x) y allows computational transfer of relations (Bremner, 12 Dec 2025).
• The quadratic relations in arity 5 among the ω_i are presented as the kernel of a rewriting morphism r: the relations are explicit combinations, and extensive computational linear algebra yields exactly 33 linearly independent quadratic relations, forming a complete presentation of 𝒟^{[2]} as a quotient of the free ternary operad FT by these relations.

This case illustrates the general mechanism by which Veronese squares encode "ternary" analogues for a given binary variety, with associated computational and combinatorial techniques required for explicit presentation (Bremner, 12 Dec 2025).

6. Ungraded Versions and the Ginzburg–Kapranov Test

"Mock" versions of Veronese powers arise in contexts where the structure is imposed on ungraded vector spaces, leading to constructions such as tCom^{n,0} with a single n-ary symmetric operation (homological degree 0), and its Koszul dual Lie^{n,0}. These exhibit surprising behavior:

• tCom^{n,0} is Koszul if and only if n is odd;
• Lie^{n,0} is Koszul if and only if n is even.

For n=3, both fail to be Koszul, yet the Ginzburg–Kapranov power series positivity test fails to detect the lack of Koszulness: the compositional inverse of the Poincaré series for tCom^{3,0} has all positive coefficients, despite the operad being non-Koszul. This highlights subtle limitations in existing Koszulness detection criteria (Dotsenko et al., 2017).

7. Context, Applications, and Computational Techniques

Veronese powers of operads yield operadic presentations of higher-arity analogues and "pure" homotopy algebra structures. Applications permeate the theory of L_infty-algebras, partially associative systems, Jordan triple systems, and the study of Koszulness for multi-ary operads. In concrete cases such as dendriform and Rota–Baxter operads, combinatorial and computational approaches—enumeration of operad monomials, construction of rewriting matrices, and lattice basis reduction (e.g., LLL algorithm)—are central for explicit presentations (Bremner, 12 Dec 2025).

A plausible implication is that further development of computational operadic techniques will be necessary to make the Veronese powers construction effective for a wider class of operads, given the combinatorial complexity of the resulting quadratic relations and the failure of direct homological improvement for most operads.