The Witt-type algebra W(a,−1)\mathcal{W}(a,-1), the semi-direct product of the Witt algebra W\mathbf{W} with its tensor density module I(a,−1)I(a,-1), is the distinguished member of the family W(a,b)\mathcal{W}(a,b): it is essentially the only one carrying a nontrivial transposed Poisson structure, i.e. a commutative associative product jointly with its Lie bracket satisfying the transposed Leibniz identity. Lubkov and Shiringovskiy (2026) began the operator structure theory of this algebra: they classified derivations, automorphisms and their local, 2-local and quasi variants, classified Rota--Baxter operators of weights 0 and 1 only under the twin assumptions of homogeneity with respect to the natural Z\mathbf{Z}-grading and invariance on the decomposition W⊕I(a,−1)\mathbf{W}\oplus I(a,-1) (block-diagonality), proved that no nontrivial odd Z2\mathbf{Z}_2-homogeneous (summand-swapping) Rota--Baxter operators of weight 1 exist, and classified Novikov--Poisson structures only in the W\mathbf{W}-compatible sense, i.e. on the Witt subalgebra. The open problem stated here asks for the completion of that classification: determine all Rota--Baxter operators of weights 0 and 1 on W(a,−1)\mathcal{W}(a,-1), on its Lie structure and on its full two-product structure, without the homogeneity and block-diagonality hypotheses, and determine all Novikov--Poisson structures on the full algebra with its given associative product, in the bracket-compatible sense of the source's Section 5, i.e. with the commutator of the Novikov product recovering the given Lie bracket. No progress on either component is known to date; even on the Witt algebra alone, the non-homogeneous Rota--Baxter classification beyond a single shifted-diagonal class remains open. For b≠−1b\neq -1, nontrivial transposed Poisson structures do not exist.

Background

The Witt algebra W\mathbf{W} is the Lie algebra of derivations of the Laurent polynomial algebra C[t,t−1]\mathbf{C}[t,t^{-1}]: it has a basis {Ln}n∈Z\{L_n\}_{n\in\mathbf{Z}} with bracket [Ln,Lm]=(n−m)Ln+m[L_n,L_m]=(n-m)L_{n+m}. For complex parameters a,ba,b, the tensor density module I(a,b)I(a,b) is the W\mathbf{W}-module with basis {Im}m∈Z\{I_m\}_{m\in\mathbf{Z}} and action [Ln,Im]=−(m+a+bn)In+m[L_n,I_m]=-(m+a+bn)I_{n+m}, and the semi-direct product W(a,b)=W⋉I(a,b)\mathcal{W}(a,b)=\mathbf{W}\ltimes I(a,b) (with [In,Im]=0[I_n,I_m]=0) is a classical family of infinite-dimensional Lie algebras of mathematical-physics origin.

A transposed Poisson algebra, introduced by C. Bai, R. Bai, L. Guo and Y. Wu, is a vector space LL carrying a commutative associative product ⋅\cdot and a Lie bracket [ , ][\,,\,] linked by the transposed Leibniz identity 2z⋅[x,y]=[z⋅x,y]+[x,z⋅y]2z\cdot[x,y]=[z\cdot x,y]+[x,z\cdot y] — the ordinary Poisson Leibniz rule with the roles of the two products exchanged. These structures arise from Novikov--Poisson algebras, defined by X. Xu: a Novikov--Poisson algebra is a triple (A,∘,⋅)(A,\circ,\cdot) in which (A,∘)(A,\circ) is a Novikov algebra (the product is left-symmetric, (x∘y)∘z−x∘(y∘z)=(y∘x)∘z−y∘(x∘z)(x\circ y)\circ z-x\circ(y\circ z)=(y\circ x)\circ z-y\circ(x\circ z), and right-commutative, (x∘y)∘z=(x∘z)∘y(x\circ y)\circ z=(x\circ z)\circ y), (A,⋅)(A,\cdot) is commutative associative, and the two products are compatible in the sense that (x⋅y)∘z=x⋅(y∘z)(x\cdot y)\circ z=x\cdot(y\circ z) and (x∘y)⋅z−x∘(y⋅z)=(y∘x)⋅z−y∘(x⋅z)(x\circ y)\cdot z-x\circ(y\cdot z)=(y\circ x)\cdot z-y\circ(x\cdot z).

Ferreira, Kaygorodov and Lopatkin proved that W(a,−1)\mathcal{W}(a,-1) carries a nontrivial transposed Poisson structure, with associative product Ln⋅Lm=Ln+mL_n\cdot L_m=L_{n+m}, Ln⋅Im=In+mL_n\cdot I_m=I_{n+m}, In⋅Im=0I_n\cdot I_m=0 alongside the Lie bracket, and that this is essentially the only interesting case: for b≠−1b\neq -1 all 12\tfrac12-derivations of W(a,b)\mathcal{W}(a,b) are trivial, and a Lie algebra with only trivial 12\tfrac12-derivations admits no nontrivial transposed Poisson structure. (12\tfrac12-derivations are the linear maps DD with D([x,y])=12([D(x),y]+[x,D(y)])D([x,y])=\tfrac12([D(x),y]+[x,D(y)]); they furnish the standard obstruction criterion for transposed Poisson structures.)

A Rota--Baxter operator of weight λ∈C\lambda\in\mathbf{C} on a Lie algebra is a linear map RR with [R(x),R(y)]=R([R(x),y]+[x,R(y)])+λR([x,y])[R(x),R(y)]=R([R(x),y]+[x,R(y)])+\lambda R([x,y]); on a transposed Poisson algebra the identity is required simultaneously for the Lie bracket and for the associative product. Since λ−1R\lambda^{-1}R has weight 1 whenever RR has nonzero weight λ\lambda, weights 0 and 1 are the decisive cases. On the Witt algebra, Gao, Liu, Bai and Jing classified the homogeneous Rota--Baxter operators of weights 0 and 1, where homogeneous of degree kk means R(Ln)⊆CLn+kR(L_n)\subseteq\mathbf{C}L_{n+k}; Tang, via the correspondence between weight-1 operators and post-Lie structures, additionally classified the operators of the shifted-diagonal shape R(Lm)=f(m)Lm+g(m)Lm+νR(L_m)=f(m)L_m+g(m)L_{m+\nu}, obtaining sixteen explicit non-homogeneous families. No classification free of such shape hypotheses is known even on W\mathbf{W} itself.

Lubkov and Shiringovskiy (2026) extended this program to the transposed Poisson algebra W(a,−1)\mathcal{W}(a,-1). They classified derivations, their local and 2-local variants (maps agreeing at each point, or at each pair of points, with some derivation), quasi-derivations, δ\delta-derivations, and the analogous automorphism groups, proving a strong rigidity picture in which all local and 2-local versions coincide with the global ones. For Rota--Baxter operators they classified: of weight 0, the operators that are simultaneously homogeneous with respect to the Z\mathbf{Z}-grading W(a,−1)=⨁n(CLn⊕CIn)\mathcal{W}(a,-1)=\bigoplus_n(\mathbf{C}L_n\oplus\mathbf{C}I_n) and block-diagonal, i.e. invariant on W⊕I(a,−1)\mathbf{W}\oplus I(a,-1) (Theorem 4.5, three families on the Lie structure); and of weight 1, the homogeneous block-diagonal operators on the full transposed Poisson structure (Theorem 4.9, with the Witt part forced by Lemma 4.11 into one of four families compatible with the associative product). They also proved that there are no nontrivial homogeneous Rota--Baxter operators of weight 1 of odd degree with respect to the Z2\mathbf{Z}_2-grading W⊕I(a,−1)\mathbf{W}\oplus I(a,-1), i.e. the summand-swapping operators with R(W)⊆I(a,−1)R(\mathbf{W})\subseteq I(a,-1) and R(I(a,−1))⊆WR(I(a,-1))\subseteq\mathbf{W} (Theorem 4.13 and Corollary 4.15). Finally, on the Novikov--Poisson side they worked in the W\mathbf{W}-compatible sense: taking the Novikov products on the Witt subalgebra classified by Kong, Chen and Bai (the graded family Ln∘Lm=(α+m)Ln+mL_n\circ L_m=(\alpha+m)L_{n+m}, α∈C\alpha\in\mathbf{C}), they verified that every member of this family, together with the associative product of W\mathbf{W}, satisfies the two Novikov--Poisson compatibility identities; by the source's Corollary 5.3 the commutator of this family is (m−n)Ln+m(m-n)L_{n+m}, the negative of the bracket [Ln,Lm]=(n−m)Ln+m[L_n,L_m]=(n-m)L_{n+m} fixed here. What these theorems leave untouched — the operators and structures without the homogeneity, block-diagonality and W\mathbf{W}-restriction hypotheses — is the subject of the present problem.

Problem Statement

The problem is to complete the operator-structure classification of the transposed Poisson algebra W(a,−1)=W⋉I(a,−1)\mathcal{W}(a,-1)=\mathbf{W}\ltimes I(a,-1) (over C\mathbf{C}, with the products Ln⋅Lm=Ln+mL_n\cdot L_m=L_{n+m}, Ln⋅Im=In+mL_n\cdot I_m=I_{n+m}, In⋅Im=0I_n\cdot I_m=0, [Ln,Lm]=(n−m)Ln+m[L_n,L_m]=(n-m)L_{n+m}, [Ln,Im]=−(m+a−n)In+m[L_n,I_m]=-(m+a-n)I_{n+m}, [In,Im]=0[I_n,I_m]=0) initiated by Lubkov and Shiringovskiy. Its two components share one scientific objective — determining which Rota--Baxter and Novikov--Poisson structures this algebra actually carries — and are jointly the completion of a single classification program:

(1) Classify all Rota--Baxter operators of weight 0 and of weight 1 on W(a,−1)\mathcal{W}(a,-1), both on its Lie structure and on its full transposed Poisson structure (where the Rota--Baxter identity must hold for the Lie bracket and for the associative product simultaneously), removing the assumptions of homogeneity with respect to the Z\mathbf{Z}-grading and of invariance on the decomposition W⊕I(a,−1)\mathbf{W}\oplus I(a,-1). In particular: does W(a,−1)\mathcal{W}(a,-1) admit any non-homogeneous Rota--Baxter operator of weight 0 or 1, or any homogeneous operator that mixes the W\mathbf{W} and I(a,−1)I(a,-1) components beyond the odd Z2\mathbf{Z}_2-homogeneous (summand-swapping) ones already proved rigid?

(2) Classify the Novikov--Poisson structures on the full algebra in the bracket-compatible sense of the source's Section 5: determine all Novikov products ∘\circ on W(a,−1)\mathcal{W}(a,-1) such that (W(a,−1),∘)(\mathcal{W}(a,-1),\circ) is a Novikov algebra, the Novikov--Poisson compatibility identities (x⋅y)∘z=x⋅(y∘z)(x\cdot y)\circ z=x\cdot(y\circ z) and (x∘y)⋅z−x∘(y⋅z)=(y∘x)⋅z−y∘(x⋅z)(x\circ y)\cdot z-x\circ(y\cdot z)=(y\circ x)\cdot z-y\circ(x\cdot z) hold with the given associative product ⋅\cdot, and the commutator x∘y−y∘xx\circ y-y\circ x recovers the given Lie bracket of W(a,−1)\mathcal{W}(a,-1); degenerate products whose commutator is not the given bracket, such as ∘≡0\circ\equiv 0, do not qualify. On the Witt subalgebra, the known graded W\mathbf{W}-compatible Novikov products whose commutator recovers the given bracket [Ln,Lm]=(n−m)Ln+m[L_n,L_m]=(n-m)L_{n+m} form the family Ln∘Lm=−(α+m)Ln+mL_n\circ L_m=-(\alpha+m)L_{n+m}, α∈C\alpha\in\mathbf{C} (the negative of the family verified in the source, whose commutator is (m−n)Ln+m(m-n)L_{n+m}; negating a product preserves the Novikov identities and both compatibility identities while reversing its commutator); in particular, decide whether this known graded family extends to the full algebra.

References

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