The Zariski cancellation problem for Poisson algebras

We propose an extension of [Formula: see text]-ary Nambu–Poisson bracket to superspace [Formula: see text] and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace [Formula: see text]. We prove in the case of the superspaces [Formula: see text] and [Formula: see text] that our [Formula: see text]-ary bracket, defined with the help of superdeterminant, satisfies the conditions for [Formula: see text]-ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of [Formula: see text]-ary bracket defined with the help of superdeterminant in the case of superspace [Formula: see text] and show that it is the sum of usual [Formula: see text]-ary Nambu–Poisson bracket and a new [Formula: see text]-ary bracket, which we call [Formula: see text]-bracket, where [Formula: see text] is the product of two odd degree smooth functions.

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Kähler-Poisson Algebras

10.3384/diss.diva-163164 ◽

2020 ◽

Author(s):

Ahmed Al-Shujary

Keyword(s):

Poisson Algebras

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A Dixmier-Moeglin equivalence for Poisson algebras with torus actions

Algebra and Its Applications - Contemporary Mathematics ◽

10.1090/conm/419/08001 ◽

2006 ◽

pp. 131-154 ◽

Cited By ~ 18

Author(s):

K. R. Goodearl

Keyword(s):

Poisson Algebras ◽

Torus Actions

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Neighborhood Reconstruction and Cancellation of Graphs

The Electronic Journal of Combinatorics ◽

10.37236/6676 ◽

2017 ◽

Vol 24 (2) ◽

Author(s):

Richard H. Hammack ◽

Cristina Mullican

Keyword(s):

Direct Product ◽

Structural Characterization ◽

Unsolved Problem ◽

Reconstruction Problem ◽

Cancellation Problem ◽

Open Vertex ◽

Isomorphic Graphs ◽

Product Of Graphs

We connect two seemingly unrelated problems in graph theory.Any graph $G$ has a neighborhood multiset $\mathscr{N}(G)= \{N(x) \mid x\in V(G)\}$ whose elements are precisely the open vertex-neighborhoods of $G$. In general there exist non-isomorphic graphs $G$ and $H$ for which $\mathscr{N}(G)=\mathscr{N}(H)$. The neighborhood reconstruction problem asks the conditions under which $G$ is uniquely reconstructible from its neighborhood multiset, that is, the conditions under which $\mathscr{N}(G)=\mathscr{N}(H)$ implies $G\cong H$. Such a graph is said to be neighborhood-reconstructible.The cancellation problem for the direct product of graphs seeks the conditions under which $G\times K\cong H\times K$ implies $G\cong H$. Lovász proved that this is indeed the case if $K$ is not bipartite. A second instance of the cancellation problem asks for conditions on $G$ that assure $G\times K\cong H\times K$ implies $G\cong H$ for any bipartite~$K$ with $E(K)\neq \emptyset$. A graph $G$ for which this is true is called a cancellation graph.We prove that the neighborhood-reconstructible graphs are precisely the cancellation graphs. We also present some new results on cancellation graphs, which have corresponding implications for neighborhood reconstruction. We are particularly interested in the (yet-unsolved) problem of finding a simple structural characterization of cancellation graphs (equivalently, neighborhood-reconstructible graphs).

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