$\delta$-Poisson and transposed $\delta$-Poisson algebras

Technical Summary of "δ-Poisson and transposed δ-Poisson algebras"

Problem Statement
The paper addresses the classification and structural analysis of two new classes of Poisson-type algebras: δ-Poisson algebras and transposed δ-Poisson algebras. These structures generalize classical Poisson algebras (where $\delta=1$) and anti-Poisson algebras (where $\delta=-1$). The study is motivated by recent investigations into shift associative algebras and the theory of $\delta$-derivations introduced by Filippov. The authors aim to determine the identities satisfied by these algebras, classify their simple instances, analyze their depolarizations (formulations in terms of a single multiplication), construct bases for their free algebras, and investigate their properties within the framework of operad theory (specifically Koszul duality and self-duality).

Methodology
The research employs a combination of algebraic manipulation, identity verification, and structural classification:

1. Identity Derivation: The authors derive fundamental identities satisfied by $\delta$-Poisson and transposed $\delta$-Poisson algebras, particularly focusing on the behavior of the Lie bracket $\{ \cdot, \cdot \}$ with respect to the commutative associative product $\cdot$.
2. Depolarization: Section 2 establishes a one-to-one correspondence between these Poisson-type algebras and specific varieties of algebras with a single multiplication. By defining the symmetric product $x \circ y = \frac{1}{2}(x \cdot y + y \cdot x)$ and the skew-symmetric product $[x, y] = \frac{1}{2}(x \cdot y - y \cdot x)$, the authors translate the two-operation definitions into single-operation identities (e.g., $f_\delta(x,y,z)=0$).
3. Classification of Simple Algebras: The paper investigates the existence of simple algebras within these varieties, utilizing ideal theory and the properties of $\delta$-derivations.
4. Free Algebra Construction: A basis for the free $\delta$-Poisson and free mixed-Poisson algebras generated by a countable set is constructed using induction and right-normed monomials, extending methods used for free Lie and free Poisson algebras.
5. Operad Theory: The authors analyze the operads governing these varieties ($\delta$-P and MP). They compute the Koszul duals, determine if the operads are Koszul, and check for self-duality using dimension calculations and Hilbert series.

Key Contributions and Results

• Structural Properties of $\delta$-Poisson Algebras:

  • For $\delta \neq 1$, the authors prove that $\delta$-Poisson algebras are almost 3-step nilpotent. Specifically, they satisfy $\{xy, zt\} = 0$ and $\{x, y\}zt = 0$.
  • Theorem 8: There are no nontrivial simple $\delta$-Poisson algebras for $\delta \neq 1$.
  • These algebras are shown to be F-manifold algebras. They satisfy the identities of Jordan brackets if and only if a specific strong identity holds.
  • The tensor product of two $\delta$-Poisson algebras yields a $\delta$-Poisson algebra.

• Structural Properties of Transposed $\delta$-Poisson Algebras:

  • Defined by the identity $x\{y, z\} = \delta(\{xy, z\} + \{y, xz\})$.
  • Unlike the standard $\delta$-Poisson case, Theorem 34 establishes that simple transposed $\delta$-Poisson algebras exist for $\delta \notin \{0, 1\}$.
  • Theorem 35: In the complex finite-dimensional case, a nontrivial simple transposed $\delta$-Poisson algebra exists only if $\delta = -1$. The authors classify these as isomorphic to two specific 3-dimensional algebras ($A_1$ and $A_2$) based on the Lie algebra $sl_2$.
  • The tensor product of two transposed $\delta$-Poisson algebras is also a transposed $\delta$-Poisson algebra.

• Depolarizations (Single Multiplication):

  • The paper provides explicit polynomial identities (e.g., $f_\delta, F_\delta, G_\delta$) that a single multiplication algebra must satisfy to generate a $\delta$-Poisson or transposed $\delta$-Poisson structure via the symmetrization/antisymmetrization process.
  • For $\delta = -1$, the algebra is shown to be anti-flexible if and only if it is shift associative.
  • For $\delta = 1$, the algebra corresponds to Kokoris algebras (noncommutative Jordan algebras).

• Free Algebras and Bases:

  • Theorem 68: A basis for the free $\delta$-Poisson algebra $\delta$-P(X) is constructed. The basis consists of three types of monomials: pure Lie monomials, products of a generator and a Lie monomial, and pure associative monomials.
  • Theorem 71: A basis for the free mixed-Poisson algebra is constructed, consisting of the union of free Lie algebra monomials and commutative associative monomials.
  • Dimensions of the $n$-th components of the operads are calculated (e.g., $\dim(\delta\text{-P}(n)) = (n-1)! + (n-2)! + 1$ for $n \geq 5$).

• Operad Theory:

  • Theorem 73: The operad $\delta$-P is self-dual for any $\delta$.
  • Theorem 75: The operad governing anti-Poisson algebras (and by extension $\delta$-P for $\delta \notin \{0, 1\}$) is not Koszul. This is demonstrated by calculating the Hilbert series and showing that the composition of the operad with its dual does not yield the identity.
  • Theorem 78: The dual of the mixed-Poisson operad is isomorphic to the free product of the Lie operad and the As-Com operad ($MP! \cong Lie * As\text{-}Com$).

Significance and Claims
The paper claims to provide a comprehensive study of these new algebraic structures, highlighting that their behavior for $\delta \neq 1$ differs significantly from classical Poisson algebras. Specifically, the non-existence of simple $\delta$-Poisson algebras contrasts with the existence of simple transposed $\delta$-Poisson algebras (specifically for $\delta = -1$). The work connects these algebras to established concepts such as F-manifolds, Jordan brackets, and shift associative algebras. The construction of bases for free algebras and the analysis of the associated operads (self-duality and non-Koszulity) are presented as fundamental contributions to the theory of non-associative algebras and operads. The authors note that the results regarding $\delta$-derivations and the classification of simple algebras offer new insights into the landscape of Poisson-type generalizations.