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$δ$-Poisson and transposed $δ$-Poisson algebras

Hani Abdelwahab, Ivan Kaygorodov, Bauyrzhan Sartayev

Abstract

We present a comprehensive study of two new Poisson-type algebras. Namely, we are working with $δ$-Poisson and transposed $δ$-Poisson algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to shift associative algebras, $F$-manifold algebras, algebras of Jordan brackets, etc. We classify simple $δ$-Poisson and transposed $δ$-Poisson algebras and found their depolarizations. We study $δ$-Poisson and mixed-Poisson algebras to be Koszul and self-dual. Bases of the free $δ$-Poisson and mixed-Poisson algebras generated by a countable set $X$ were constructed.

$δ$-Poisson and transposed $δ$-Poisson algebras

Abstract

We present a comprehensive study of two new Poisson-type algebras. Namely, we are working with -Poisson and transposed -Poisson algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to shift associative algebras, -manifold algebras, algebras of Jordan brackets, etc. We classify simple -Poisson and transposed -Poisson algebras and found their depolarizations. We study -Poisson and mixed-Poisson algebras to be Koszul and self-dual. Bases of the free -Poisson and mixed-Poisson algebras generated by a countable set were constructed.

Paper Structure

This paper contains 15 sections, 54 theorems, 60 equations.

Table of Contents

  1. New Poisson-type algebras
  2. $\delta$-Poisson algebras
  3. Transposed $\delta$-Poisson algebras
  4. Poisson-type algebras in one multiplication
  5. $\delta$-Poisson algebras in one multiplication
  6. Scalar-Poisson algebras in one multiplication
  7. Transposed $\delta$-Poisson algebras in one multiplication
  8. Transposed scalar-Poisson algebras in one multiplication
  9. Mixed-Poisson algebras in one multiplication
  10. Free algebras
  11. Free $\delta$-Poisson algebras
  12. Free mixed-Poisson algebras
  13. Operads
  14. Dual operad of the $\delta$-Poisson operad
  15. Mixed-Poisson operad

Key Result

Theorem 2

Let $(\mathcal{A},\cdot)$ be a shift associative algebra, i.e. an algebra satisfying the identity $(xy)z=y(zx)$. Then $(\mathcal{A}, \circ, [\cdot,\cdot])$ is anti-Poisson-Jordan; $(\mathcal{A}, \circ, [\cdot,\cdot])$ is an anti-Poisson algebra if and only if $(\mathcal{A},\cdot)$ satisfies the iden

Theorems & Definitions (115)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 6
  • proof
  • Corollary 7
  • Theorem 8
  • proof
  • ...and 105 more