Transposed Poisson Algebra

• Transposed Poisson algebra is a structure combining a commutative associative product and a Lie bracket where multiplication acts as a 1/2-derivation on the bracket.
• It establishes a dual relationship to classical Poisson algebras and connects deeply with weak Leibniz and Novikov-Poisson structures through polarization techniques.
• The framework supports comprehensive classifications across Lie families, deformation quantization, and tensor-product constructions, influencing operadic and noncommutative generalizations.

A transposed Poisson algebra is a vector space $A$ equipped with a commutative associative product $\cdot$ and a Lie bracket $[\ ,\ ]$ such that

$$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$

This compatibility is the transposed analogue of the usual Poisson Leibniz rule: in an ordinary Poisson algebra the bracket acts as a derivation of the product, whereas here multiplication by $z$ acts on the Lie bracket as a $\tfrac12$-derivation. The notion was introduced as a dual version of Poisson algebra and has since been connected with weak Leibniz algebras, Novikov-Poisson algebras, nilpotency theory, operadic structures, quantization by Novikov deformations, and noncommutative double analogues (Bai et al., 2020, AskarDzhumadil'daev, 2023, Jin et al., 28 Apr 2026, Fairon et al., 1 Jul 2026).

1. Definition and basic viewpoints

The defining data of a transposed Poisson algebra are standard on each side: $(A,\cdot)$ is a commutative associative algebra, $(A,[\ ,\ ])$ is a Lie algebra, and the mixed identity is

$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$

Many papers formulate this as a statement about multiplication operators. If $M_z(x)=z\cdot x$, then the compatibility is equivalent to requiring that every $\cdot$0 be a $\cdot$1-derivation of the Lie algebra: $\cdot$2 In finite-dimensional structure theory, the same idea is encoded by operators $\cdot$3 and $\cdot$4, with the equivalent identity

$\cdot$5

This operator form is central in Engel-type arguments and in classifications by $\cdot$6-derivations (Kaygorodov et al., 2022, Jin et al., 28 Apr 2026).

The literature also treats transposed Poisson algebras as a special kind of dialgebra: a vector space with two binary operations. In that language, a subalgebra $\cdot$7 satisfies $\cdot$8, and an ideal $\cdot$9 satisfies $[\ ,\ ]$0. Some works call the structure trivial when either the associative product or the Lie bracket is trivial. Characteristic restrictions depend on the context: the general dialgebra definition is usually stated over fields of characteristic $[\ ,\ ]$1, while polarization arguments impose additional restrictions (Jin et al., 28 Apr 2026, AskarDzhumadil'daev, 2023).

2. Polarization and the weak Leibniz origin

A major structural characterization identifies transposed Poisson algebras as the polarized form of weak Leibniz algebras. A weak Leibniz algebra is a one-product nonassociative algebra $[\ ,\ ]$2 satisfying

$[\ ,\ ]$3

where $[\ ,\ ]$4. No associativity, commutativity, anti-commutativity, or Lie identity is assumed for the single product. Every Leibniz algebra is weak Leibniz, but not conversely (AskarDzhumadil'daev, 2023).

The passage to transposed Poisson structure is by polarization. If $[\ ,\ ]$5 is a one-product algebra, set

$[\ ,\ ]$6

Conversely, if $[\ ,\ ]$7 is a dialgebra, its depolarization is the one-product algebra with

$[\ ,\ ]$8

The main equivalence theorem states that if $[\ ,\ ]$9 is weak Leibniz over a field of characteristic $$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$0, then its polarization is a transposed Poisson algebra; conversely, if $$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$1 is transposed Poisson over a field of characteristic $$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$2, then its depolarization is weak Leibniz. Hence, in characteristic $$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$3, an algebra is weak Leibniz if and only if its polarization is transposed Poisson (AskarDzhumadil'daev, 2023).

This characterization explains the internal decomposition of the weak Leibniz product. The skew part becomes a Lie bracket, the symmetric part becomes a commutative associative product, and the weak Leibniz identities polarize to

$$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$4

The same paper also proves that the weak Leibniz operad is self-dual and not Koszul. Since transposed Poisson algebras arise from polarization of weak Leibniz algebras, this operadic behavior informs the one-product operadic background of the theory (AskarDzhumadil'daev, 2023).

3. Standard constructions and sources

The basic explicit construction starts from a commutative associative algebra $$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$5 with a derivation $$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$6. Then

$$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$7

defines a Lie bracket, and $$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$8 is a transposed Poisson algebra. In polynomial algebras this yields brackets of the form

$$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y] \qquad \text{for all }x,y,z\in A.$$9

with $z$0 (Bai et al., 2020).

A second major source is Novikov-Poisson theory. If $z$1 is a Novikov-Poisson algebra, then the commutator

$z$2

turns $z$3 into a transposed Poisson algebra. More generally, transposed Novikov-Poisson algebras were introduced so that their affinization $z$4 is transposed Poisson. For a transposed Novikov-Poisson algebra $z$5, the loop algebra carries

$z$6

$z$7

and this is transposed Poisson if and only if $z$8 satisfies the transposed Novikov-Poisson identities. The same paper gives a tensor-product construction with a right differential Novikov-Poisson algebra and proves that tensor products of transposed Novikov-Poisson algebras remain in the same class (Jin et al., 13 Feb 2026).

Transposed Poisson algebras are also closed under tensor products in the purely transposed Poisson category. If $z$9 and $\tfrac12$0 are transposed Poisson algebras, then $\tfrac12$1 becomes one under

$\tfrac12$2

$\tfrac12$3

In the nilpotent/solvable setting, tensor products preserve nilpotency and solvability when one factor has the corresponding property (Bai et al., 2020, Jin et al., 28 Apr 2026).

Quantization enters through Novikov deformations. If a commutative associative algebra is deformed into a Novikov algebra $\tfrac12$4, then the classical limit bracket

$\tfrac12$5

is transposed Poisson. All transposed Poisson algebras of Novikov-Poisson type, including all unital transposed Poisson algebras, are quantizable in this sense (Chen et al., 2024).

4. Classifications on major Lie families

Several concrete Lie algebras admit explicit classifications of their transposed Poisson structures. The results are sharply rigidity-sensitive: some families admit only mutations or Poisson-type structures, while others allow one or a few exceptional non-Poisson classes.

Lie algebra family	Classification pattern	Source
Witt type $\tfrac12$6	Mutations of the group algebra product when $\tfrac12$7; three-summand or piecewise mutation forms when $\tfrac12$8 or $\tfrac12$9	(Kaygorodov et al., 2022)
Generalized Witt $(A,\cdot)$0	Trivial if $(A,\cdot)$1; mutations of the group algebra structure on $(A,\cdot)$2 if $(A,\cdot)$3	(Kaygorodov et al., 2023)
Block $(A,\cdot)$4	In bijection with commutative associative products on a complement of $(A,\cdot)$5 with values in $(A,\cdot)$6; all are usual Poisson structures	(Kaygorodov et al., 2023)
Block $(A,\cdot)$7	Trivial for $(A,\cdot)$8; one nontrivial isomorphism class for $(A,\cdot)$9	(Kaygorodov et al., 2022)
Upper triangular $(A,[\ ,\ ])$0	For $(A,[\ ,\ ])$1: Poisson type, or orthogonal sum with a fixed non-Poisson structure; for $(A,[\ ,\ ])$2: one extra class	(Kaygorodov et al., 2023)
Full matrix $(A,[\ ,\ ])$3	Up to isomorphism, one non-trivial structure, of Poisson type	(Kaygorodov et al., 2023)
Incidence Lie algebra $(A,[\ ,\ ])$4	Sum of a Poisson-type part, a mutational part, and a $(A,[\ ,\ ])$5-part supported on extreme pairs $(A,[\ ,\ ])$6	(Kaygorodov et al., 2023)

For Witt type Lie algebras $(A,[\ ,\ ])$7, the classification is controlled by $(A,[\ ,\ ])$8-derivations. When $(A,[\ ,\ ])$9, all transposed Poisson products are mutations of the group algebra multiplication $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$0. When $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$1, the algebra decomposes into three coset summands, each carrying a mutated product and with zero cross-multiplication. When $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$2, the product has a piecewise form depending on whether the basis indices lie in $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$3 (Kaygorodov et al., 2022).

For the Lie algebra of upper triangular matrices $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$4, the central classification is especially explicit. For $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$5, every transposed Poisson structure is either of Poisson type or the orthogonal sum of a Poisson-type structure with the fixed multiplication

$2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$6

For $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$7, there is one more class. The same paper proves that $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$8, up to isomorphism, admits only one non-trivial transposed Poisson structure, and it is of Poisson type (Kaygorodov et al., 2023).

For Lie incidence algebras $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y].$9, the classification is combinatorial. Every transposed Poisson structure is the sum of a Poisson-type structure, a mutational structure, and a $M_z(x)=z\cdot x$0-structure determined by a map $M_z(x)=z\cdot x$1, where $M_z(x)=z\cdot x$2 consists of maximal comparable pairs not contained in cycles. The underlying $M_z(x)=z\cdot x$3-derivations decompose into central-valued, inner, and chain/cycle-constant pieces (Kaygorodov et al., 2023).

5. Nilpotency, radicals, and Frattini theory

The finite-dimensional structure theory of transposed Poisson algebras has been developed in detail. For a subalgebra $M_z(x)=z\cdot x$4, the lower central series is defined by

$M_z(x)=z\cdot x$5

and Proposition 2.4 simplifies this to

$M_z(x)=z\cdot x$6

The derived series is

$M_z(x)=z\cdot x$7

These definitions support analogues of Lie- and associative-algebra nilpotency and solvability (Jin et al., 28 Apr 2026).

The main structural theorem is an Engel-type criterion: $M_z(x)=z\cdot x$8 In finite dimension, this is equivalent to the nilpotency of all left multiplication operators $M_z(x)=z\cdot x$9 and $\cdot$00. The proof uses the operator identity

$\cdot$01

to reorder words in the operators and force vanishing from the nilpotency of the associative and Lie sides (Jin et al., 28 Apr 2026).

The same paper proves that if $\cdot$02 is finite-dimensional and Lie nilpotent, then the derived Lie algebra $\cdot$03 is a nilpotent ideal. From this it deduces that the nilpotent radical $\cdot$04 coincides with the associative radical $\cdot$05 in the Lie-nilpotent case. It also develops a Frattini theory: for finite-dimensional $\cdot$06,

$\cdot$07

the Frattini ideal $\cdot$08 is associative nilpotent, and in the nilpotent case

$\cdot$09

If $\cdot$10, then the zero socle coincides with the nilpotent radical, and in the Lie-nilpotent case one obtains decompositions

$\cdot$11

and

$\cdot$12

These results show that transposed Poisson nilpotency behaves neither as a mere copy of Lie theory nor of commutative associative theory, but through a coupled dialgebra structure (Jin et al., 28 Apr 2026).

6. Operadic, GD, and splitting perspectives

Transposed Poisson algebras admit a precise reformulation in the language of Gel'fand-Dorfman theory. The variety of transposed Poisson algebras coincides with the variety of Gelfand-Dorfman algebras in which the Novikov multiplication is commutative. In that form, the transposed Poisson identity can be written as

$\cdot$13

The corresponding shuffle-operad Gröbner-Shirshov basis has been computed up to degree $\cdot$14, and the same work proves that every transposed Poisson algebra is an $\cdot$15-manifold. It also verifies that the known special identities of GD-algebras hold in transposed Poisson algebras and formulates the conjecture that every transposed Poisson algebra is special, meaning embeddable into a differential Poisson algebra (Sartayev, 2023).

A different line of work studies splittings of the two operations. Because a transposed Poisson algebra contains both a commutative associative product and a Lie bracket, each can be left unsplit, classically split, or second-split. This yields mixed families interpreted by representations of the transposed Poisson algebra on the space itself and on the dual space. The paper constructs $\cdot$16 structures on the underlying space and another $\cdot$17 on the dual, including TZPO algebras, which are described as “pre-transposed Poisson algebras” and are viewed operadically as successors of the transposed Poisson operad. A decisive difference from the ordinary Poisson case is that for a representation $\cdot$18 of a transposed Poisson algebra, $\cdot$19 is not automatically again a representation; this failure is precisely why the theory splits into distinct “on the space” and “on the dual space” branches (Liu et al., 2023).

7. Generalizations and current directions

Several recent extensions broaden the notion far beyond the original commutative-Lie setting. In the super case, a transposed Poisson superalgebra consists of a commutative associative superalgebra, a Lie superalgebra, and the graded compatibility

$\cdot$20

Even derivations of commutative superalgebras produce genuine transposed Poisson superalgebras, while odd derivations do not fit the usual parity requirements and instead lead to a new TP-compatible supermodule/Jordan-type structure. The same paper shows that a transposed Poisson superalgebra with an even derivation yields a $\cdot$21-Lie superalgebra (Abramov et al., 21 Mar 2025).

Parameterized generalizations also exist. Transposed $\cdot$22-Poisson algebras replace the coefficient $\cdot$23 by $\cdot$24: $\cdot$25 On the null-filiform associative algebra $\cdot$26, the structure theory changes according to the roots of

$\cdot$27

with $\cdot$28 recovering ordinary transposed Poisson algebras. The same paper proves that ordinary $\cdot$29-Poisson algebra structures on these null-filiform associative algebras are all trivial (Daukeyeva et al., 7 Jun 2025).

The quantization problem has a specifically transposed form. Novikov deformations of commutative associative algebras have classical limits that belong to a subclass of transposed Poisson algebras, called those of quantizable type. All transposed Poisson algebras of Novikov-Poisson type, including all unital transposed Poisson algebras, can be quantized; the paper also classifies, up to equivalence, the quantizations of $\cdot$30-dimensional complex transposed Poisson algebras with non-abelian Lie brackets (Chen et al., 2024).

Two further expansions move in different directions. First, the noncommutative analogue is the double transposed Poisson algebra: on a unital associative algebra $\cdot$31, every such structure is governed by a single derivation

$\cdot$32

and it induces a $\cdot$33-equivariant transposed Poisson structure on each representation algebra $\cdot$34. The same paper introduces $\cdot$35-transposed Poisson structures and uses the trace map to obtain transposed Poisson structures on $\cdot$36 (Fairon et al., 1 Jul 2026). Second, Hopf-module theory has been adapted to the transposed setting: if $\cdot$37 is a Hopf algebra and $\cdot$38 an $\cdot$39-comodule transposed Poisson algebra, then under a right $\cdot$40-colinear algebra map $\cdot$41 into the transposed Poisson center, the fundamental theorem of transposed Poisson $\cdot$42-Hopf modules reconstructs every module as

$\cdot$43

and yields relative projectivity in the corresponding module category (Ning et al., 10 Sep 2025).

These developments indicate a stable pattern. Transposed Poisson algebra is no longer only a dualized Leibniz rule on a commutative algebra with Lie bracket; it is a node linking weak Leibniz polarization, Novikov and Gel'fand-Dorfman structures, radical and Frattini theory, representation-scheme geometry, and several graded and noncommutative extensions.