Kolesnikov (2017) developed Gröbner–Shirshov basis (GSB) theory for pre-associative (dendriform) algebras and posed three open problems: find the GSB of the universal pre-associative envelope U(−)(L)U_{(-)}(L) of a pre-Lie algebra LL; decide whether the triple (preAs,preLie,(−))(\mathrm{preAs},\mathrm{preLie},(-)) has the Poincaré–Birkhoff–Witt (PBW) property; and find the GSB of Uψ(B)U_\psi(B) for a brace algebra BB, where ψ:Brace→preAs\psi:\mathrm{Brace}\to\mathrm{preAs} is Chapoton's operad morphism. Later literature resolves the first two: Dotsenko–Tamaroff (2021) proved that (preLie,preAs)(\mathrm{preLie},\mathrm{preAs}) is a PBW pair, and Gubarev (2020a, 2020b) proved the embedding of every pre-Lie algebra into its universal pre-associative envelope and constructed that envelope with an explicit linear basis via Gröbner–Shirshov bases for associative Rota–Baxter algebras. The surviving open core is the third problem: compute an explicit Gröbner–Shirshov basis, in the sense of Kolesnikov's Composition–Diamond lemma for pre-associative algebras, of the universal pre-associative envelope Uψ(B)U_\psi(B) of an arbitrary brace algebra BB, thereby obtaining irreducible words as a linear basis and normal forms. The PBW property for (preAs,Brace,ψ)(\mathrm{preAs},\mathrm{Brace},\psi) is already known (Chapoton 2002), so the task is the explicit combinatorial reduction system, for which no Gröbner–Shirshov basis is known.

Background

A pre-associative algebra (usually called a dendriform algebra, introduced by Loday) is a vector space with two bilinear operations ≻\succ and ≺\prec satisfying the three identities x≻(y≻z)=(x≻y)≻z+(x≺y)≻z,x≻(y≺z)=(x≻y)≺z,x≺(y≻z)+x≺(y≺z)=(x≺y)≺z,x\succ(y\succ z)=(x\succ y)\succ z+(x\prec y)\succ z,\quad x\succ(y\prec z)=(x\succ y)\prec z,\quad x\prec(y\succ z)+x\prec(y\prec z)=(x\prec y)\prec z, which together imply that the sum x∗y=x≻y+x≺yx\ast y=x\succ y+x\prec y is an associative product; dendriform algebras thus 'split' associativity into two pieces. A pre-Lie algebra (also called left-symmetric or Vinberg algebra) is a vector space with a product satisfying (xy)z−x(yz)=(yx)z−y(xz)(xy)z-x(yz)=(yx)z-y(xz); every pre-associative algebra is pre-Lie under x⋅y=x≻y−y≺xx\cdot y=x\succ y-y\prec x. A brace algebra is a vector space with a family of operations x{y1,…,yn}x\{y_1,\dots,y_n\}, n≥1n\ge 1, introduced by Gerstenhaber and Voronov as the higher (brace) operations on the Hochschild complex (the standard cochain complex of an associative algebra), satisfying quadratic identities that expand a brace applied to a brace: (z{x1,…,xn}){y1,…,ym}=∑z{Y0, x1{Y1}, Y2, …, xn{Y2n−1}, Y2n},(z\{x_1,\dots,x_n\})\{y_1,\dots,y_m\}=\sum z\{Y_0,\,x_1\{Y_1\},\,Y_2,\,\dots,\,x_n\{Y_{2n-1}\},\,Y_{2n}\}, where the sum runs over all decompositions of the ordered tuple (y1,…,ym)(y_1,\dots,y_m) into 2n+12n+1 successive (possibly empty) intervals Y0,…,Y2nY_0,\dots,Y_{2n}, and x{ }x\{\ \} with an empty family is read as xx; the binary brace x{y}x\{y\} is a pre-Lie product. Chapoton (2002) proved that the category of connected dendriform bialgebras (dendriform Hopf algebras) is equivalent to the category of brace algebras, and, en route, constructed an explicit morphism of operads ψ:Brace→preAs\psi:\mathrm{Brace}\to\mathrm{preAs} (operads are devices encoding algebraic structures of a given type; a morphism of operads ω:W→V\omega:W\to V turns every VV-algebra into a WW-algebra, and every WW-algebra AA then has a universal enveloping VV-algebra Uω(A)U_\omega(A), generated by the image of a canonical linear map A→Uω(A)A\to U_\omega(A) and carrying a natural ascending filtration). On planar corollas (the generators of the brace operad) the morphism is given by the explicit alternating formula ψ(Corln)(x1,…,xn+1)=∑i=1n+1(−1)i γ(x2,…,xi)≻x1≺δ(xi+1,…,xn+1),\psi(\mathrm{Corl}_n)(x_1,\dots,x_{n+1})=\sum_{i=1}^{n+1}(-1)^i\,\gamma(x_2,\dots,x_i)\succ x_1\prec\delta(x_{i+1},\dots,x_{n+1}), where γ\gamma and δ\delta are the iterated dendriform products defined by γ(x1)=δ(x1)=x1\gamma(x_1)=\delta(x_1)=x_1, γ(x1,…,xn)=x1≺γ(x2,…,xn)\gamma(x_1,\dots,x_n)=x_1\prec\gamma(x_2,\dots,x_n) (left-nested ≺\prec-powers), and δ(x1,…,xn)=δ(x1,…,xn−1)≻xn\delta(x_1,\dots,x_n)=\delta(x_1,\dots,x_{n-1})\succ x_n (right-nested ≻\succ-powers), with the endpoint terms read as x1≺δ(x2,…,xn+1)x_1\prec\delta(x_2,\dots,x_{n+1}) for i=1i=1 and γ(x2,…,xn+1)≻x1\gamma(x_2,\dots,x_{n+1})\succ x_1 for i=n+1i=n+1. Following Mikhalev and Shestakov, a triple (V,W,ω)(V,W,\omega) as above is said to have the Poincaré–Birkhoff–Witt (PBW) property if the associated graded algebra gr Uω(A)\mathrm{gr}\,U_\omega(A) is isomorphic, as a VV-algebra, to Uω(A(0))U_\omega(A^{(0)}) for every WW-algebra AA, where A(0)A^{(0)} is the WW-algebra on the same underlying space with zero operations; it is known (Chapoton 2002) that (preAs,Brace,ψ)(\mathrm{preAs},\mathrm{Brace},\psi) has the PBW property. A Gröbner–Shirshov basis (GSB) is a system of defining relations of a presented algebra, closed under completion, such that every 'composition' — every overlap or inclusion ambiguity between the leading monomials of two relations — reduces to zero modulo the system; the Composition–Diamond lemma (the non-commutative, non-associative analogue of Buchberger's criterion) then guarantees that the words irreducible with respect to the system form a linear basis of the presented algebra, which is the standard route to PBW-type statements. Bokut, Chen, and Li (2010) developed this method for pre-Lie (right-symmetric) algebras and, as an application, showed that the initial defining relations of the universal enveloping pre-Lie algebra of an arbitrary Lie algebra already form a GSB; Kolesnikov (2017) developed the same theory for pre-associative algebras and, as an application, computed the GSB of the universal enveloping pre-associative algebra of an arbitrary associative algebra, establishing the PBW property for (preAs,As,(∗))(\mathrm{preAs},\mathrm{As},(\ast)).

Problem Statement

Let BB be a brace algebra over a field, with a well-ordered basis XX and brace operations x{y1,…,yn}x\{y_1,\dots,y_n\} given by structure constants. Let Uψ(B)=preAs⟨X∣S⟩U_\psi(B)=\mathrm{preAs}\langle X\mid S\rangle be the universal pre-associative (dendriform) envelope of BB relative to Chapoton's operad morphism ψ:Brace→preAs\psi:\mathrm{Brace}\to\mathrm{preAs}, presented by generators XX and defining relations SS stating, for all x,y1,…,yn∈Xx,y_1,\dots,y_n\in X and n≥1n\ge 1, that the brace operation x{y1,…,yn}x\{y_1,\dots,y_n\} (expanded in the basis XX) coincides with the dendriform polynomial obtained by applying ψ\psi to the corresponding brace monomial via the explicit corolla formula quoted in the Background. Compute an explicit Gröbner–Shirshov basis of this presentation inside the free pre-associative algebra preAs⟨X⟩\mathrm{preAs}\langle X\rangle, in the sense of Kolesnikov's Composition–Diamond lemma for pre-associative algebras and relative to an admissible monomial well-order on dendriform words extending the well-order on XX, which the solver specifies; the Gröbner–Shirshov basis is to be given explicitly, as a finite set or as a recursive scheme, with its irreducible words forming a linear basis of Uψ(B)U_\psi(B) and inducing the normal-form reduction system. In particular, decide whether the initial relations SS themselves form a Gröbner–Shirshov basis for a suitable ordering of this kind (as happens in the analogous (preLie,Lie,(−))(\mathrm{preLie},\mathrm{Lie},(-)) case, where the initial relations of the universal enveloping pre-Lie algebra of a Lie algebra are already a Gröbner–Shirshov basis).

References

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