Non-Degenerate Shifted Poisson Structures

• Non-degenerate shifted Poisson structures are higher analogues of classical symplectic forms that create an isomorphism between shifted cotangent and tangent complexes.
• They are defined using dg-Lie algebra techniques and the Maurer–Cartan formalism, ensuring robust non-degeneracy in derived algebraic and non-commutative settings.
• Applications span deformation quantization, representation theory, and moduli space analysis, with symplectic realizations linking Poisson structures to higher geometric frameworks.

Non-degenerate shifted Poisson structures form the foundation for the study of higher Poisson geometry within the context of derived algebraic, differential, and non-commutative geometry. These structures generalize the notion of symplectic and Poisson brackets to the field of derived stacks, dg-manifolds, and commutative and non-commutative dg-algebras, often parameterized by a cohomological degree shift $n$. The non-degeneracy condition identifies those shifted Poisson structures whose underlying bivector field induces an isomorphism between the cotangent and (appropriately shifted) tangent complexes, making them the higher analogues of symplectic forms. There is a canonical equivalence between non-degenerate $n$-shifted Poisson and $n$-shifted symplectic structures, and this identification is central to deformation quantization, representation theory, and the global study of moduli and critical loci.

1. Definition of Shifted Poisson Structures

Given a commutative dg-algebra $A$ (or more generally a derived Artin stack $X$), the space of $n$-shifted Poisson structures is

$$\text{Pois}(A,n) = \operatorname{fib}\left(\text{Alg}_{P_{n+1}} \to \text{Alg}_{\text{Comm}}\right)$$

over $A$, where $P_{n+1}$ is the $(n+1)$-shifted Poisson operad. Concretely, for a derived Artin stack $X$, one forms the graded dg-Lie algebra of shifted polyvectors: $$\mathrm{Pol}(X,n) = \Gamma\left(X_{\mathrm{dR}}, \operatorname{Hom}\left(\mathbb{L}_X[-n-1]^{\otimes}, \mathcal{O}_X\right)\right)$$ with the Schouten bracket $[\cdot, \cdot]$ of degree $-n-1$.

An $n$-shifted Poisson structure $\pi$ is a weight $\geq 2$ polyvector of total cohomological degree $n+2$ satisfying the Maurer–Cartan equation

$[\pi, \pi] = 0$

in $\mathrm{Pol}(X,n)$. These structures can be equivalently described in terms of filtered dg-Lie algebras $F^2\mathrm{Pol}(X,n)[n+1]$, or via Cartan or Chevalley-Eilenberg models for stacks with symmetry.

For $A$ a commutative dg-algebra presented as $\mathrm{Sym}(V)$ with Maurer–Cartan deformation $a \in \operatorname{Der}^1(A)$, the space of $n$-shifted polyvectors is given by

$$\mathrm{Pol}(A, n) = \prod_{m \geq 0} \mathrm{Sym}^m_A(T_A[-n-1])$$

with the shifted Schouten–Nijenhuis bracket.

2. Non-degeneracy and Equivalence with Shifted Symplectic Forms

Given a representative $\pi \in \operatorname{Pol}^2(X, n)$, the non-degeneracy condition is imposed via the contraction map

$\pi^\sharp : \mathbb{L}_X \to \mathbb{T}_X[-n]$

where $\mathbb{L}_X$ and $\mathbb{T}_X$ are the cotangent and tangent complexes of $X$, respectively. $\pi$ is called non-degenerate if $\pi^\sharp$ is a quasi-isomorphism of perfect complexes. This criterion can be adapted to Cartan and Chevalley-Eilenberg settings, and to the non-commutative context, where $A$ is a dg-associative algebra and the Kähler differentials and tangent bimodules are replaced by their non-commutative analogues.

The equivalence theorem—originally established for derived Artin stacks but now proven in a variety of settings (algebraic, differential, analytic, and even non-commutative)—is that the space of non-degenerate $n$-shifted Poisson structures is weakly equivalent to the space of $n$-shifted symplectic forms: $$\text{Pois}^{\mathrm{nd}}(X, n) \simeq \text{Symp}(X, n)$$ In this equivalence, the inverse 2-form $\omega$ satisfies $\omega^\sharp: \mathbb{T}_X \to \mathbb{L}_X[n]$ with $\omega^\sharp = (\pi^\sharp)^{-1}$. The equivalence is constructed at the level of (semi-)simplicial sets, managing higher coherences via obstruction theory in nilpotent dg-Lie algebras and leveraging the formalism of the Maurer–Cartan space.

3. Symplectic Realizations and Lagrangian Correspondences

A foundational principle is that every (non-degenerate) shifted Poisson structure appears as the coisotropic data induced on a Lagrangian morphism to a higher-shifted symplectic stack. Specifically, if $L$ carries an $(n-1)$-shifted Poisson structure, a symplectic realization is given by an $n$-shifted Lagrangian $f: L \to X$ into an $n$-shifted symplectic stack $X$ such that pulling back the symplectic structure recovers the original Poisson bracket on $L$.

A particularly important class comprises symplectic realizations arising from Manin pairs and Manin triples. For a Manin pair $(\mathfrak{g} \subset \mathfrak{d})$ with a non-degenerate invariant bilinear form, the map $BG \to BD$ realizes the 1-shifted Poisson structure on $BG$ as a symplectic realization of the quasi-Poisson structure. Likewise, for a Manin triple $(\mathfrak{d}, \mathfrak{g}, \mathfrak{g}^*)$, the Lu–Weinstein groupoid arises as a symplectic realization via a 2-shifted Lagrangian correspondence, knitting together the representation theory and geometry of quantum groups (Safronov, 2017).

The general pattern observed in all examples is:

• Start from a non-degenerate $n$-shifted Poisson structure,
• Exhibit an $(n+1)$-shifted symplectic stack $Y$ and a Lagrangian $f: L \to Y$,
• The induced coisotropic structure on $L$ recovers the original Poisson structure, with non-degeneracy equating to the invertibility of $\pi^\sharp$ and its correspondence with $\omega^\sharp$.

4. Concrete Models and Key Examples

Commutative and Chevalley-Eilenberg Algebras

For the Chevalley-Eilenberg algebra $\mathrm{CE}(\mathfrak{g})$ of an ordinary Lie algebra:

• For $n=1$, $1$-shifted Poisson structures correspond to quasi-Lie bialgebra structures—equivalently, a cobracket $\delta$ and a 3-cocycle $\varphi$ satisfying compatibility and co-Jacobi identities. Non-degeneracy is tantamount to invertibility of $\delta$, i.e., $\pi^\sharp$ being a quasi-isomorphism (Kemp et al., 17 Dec 2024).
• For $n=2$, they correspond to $G$-invariant symmetric bilinear forms $B \in (\mathrm{Sym}^2 \mathfrak{g})^G$, with non-degeneracy requiring $B$ to be non-degenerate.

Generalizations to Lie 2-algebras enumerate shifted Poisson structures for $n \in \{1,2,3,4\}$, with non-degeneracy deduced from the invertibility of appropriate weight-2 components.

Global Quotients and the Cartan Model

For quotients $[\operatorname{Spec} A / G]$ by a reductive group, Cartan models for shifted polyvectors and de Rham forms yield explicit Maurer–Cartan conditions and non-degeneracy criteria in terms of the cohomology of the Cartan tangent complex and perfectness of the cotangent structure. The leading term $\pi_2$ induces an isomorphism between the shifted cotangent and tangent complexes (Yeung, 2021).

Moduli Spaces

The derived moduli stack of complexes over a Calabi-Yau variety of dimension $d$ possesses a $(1-d)$-shifted Poisson structure, and in the elliptic ($d=1$) case, 0-shifted Poisson structures naturally foliate the moduli stack by symplectic substacks (Hua et al., 2017).

Non-commutative Setting

Non-degenerate $n$-shifted double Poisson structures on dg-associative algebras are defined as Maurer–Cartan elements of a dg-Lie algebra of non-commutative polyvectors. The non-degeneracy condition is that the induced map from Kähler differentials (modulo commutators, shifted by $-n$) to the adjoint representation bimodule is a quasi-isomorphism. There is a canonical equivalence between spaces of non-degenerate $n$-shifted double Poisson and $n$-shifted bisymplectic structures (Pridham, 2020). These induce, under representation functors and commutative quotients, the classical theory of shifted Poisson and symplectic structures.

Table: Key Models and Their Non-degeneracy Criteria

Model	Polyvector space / algebraic data	Non-degeneracy condition
Derived Artin stack $X$ (CPTVV/Pridham)	$\mathrm{Pol}(X,n)$	$$\pi_2^\sharp: \mathbb{L}_X \rightarrow \mathbb{T}_X[-n]$$ q-iso
Chevalley-Eilenberg algebra $\mathrm{CE}(\mathfrak{g})$	$(\delta,\varphi)$ for $n=1$; $B \in (\mathrm{Sym}^2\mathfrak{g})^G$ for $n=2$	Invertibility of $\delta$ or $B$
Global quotient $[\operatorname{Spec} A/G]$	Cartan model polyvectors $\mathrm{Pol}_{\mathrm{Car}}(X,n)$	$$\pi_2^\sharp: \Omega^1_{\mathrm{Car}}(A)[n] \xrightarrow{\sim} \mathcal{T}_{\mathrm{Car}}(A)$$ q-iso
Non-commutative dg-algebras	Non-commutative polyvectors $\widehat{\mathrm{Pol}}^{\mathrm{nc}}(A,n)$	$\pi_2^\sharp$ quasi-isomorphism of bimodules

5. Deformation Quantization, Vanishing Cycles, and Applications

For non-degenerate shifted Poisson structures—especially those admitting a compatible formal derivation—there exists a unique self-dual deformation quantization to a BD$_n$-algebra (e.g., filtered quantization compatible with involution $\hbar \mapsto -\hbar$). In exact settings, the classification of quantizations and the correspondence with symplectic geometry is canonical and involves no ambiguity arising from choices of associators, generalizing Fedosov's construction to the derived and higher settings (Pridham, 10 Nov 2025).

A central application is to the study of the derived critical locus $RCrit(Y, f)$ of a function $f$ on a smooth manifold or analytic space $Y$, which possesses a canonical exact $(-1)$-shifted symplectic structure, and hence a non-degenerate $(-1)$-shifted Poisson structure. Its deformation quantization produces, via Riemann–Hilbert correspondence, the perverse sheaf of vanishing cycles equipped with monodromy, relating global microlocal sheaf theory and deformation quantization.

Further examples include symplectic groupoids integrating Poisson manifolds, Feigin–Odesskii moduli spaces, higher moduli of local systems, and moduli of perfect complexes, demonstrating that non-degenerate shifted Poisson geometry is deeply intertwined with representation theory, topological field theory, and derived algebraic geometry.

6. Extensions and Non-commutative Generalizations

Non-degenerate shifted Poisson theory extends beyond commutative geometry. Non-commutative analogues involve shifted double Poisson and bisymplectic structures (Pridham, 2020). For a cofibrant dg-associative algebra $A$, $n$-shifted double Poisson structures are Maurer–Cartan elements in a completed complex of multi-derivations, with non-degeneracy encoded in the invertibility of the binary operation on the Kähler differentials. There exists a canonical equivalence between the spaces of non-degenerate double Poisson and bisymplectic structures. Under passage to commutative quotients or via representation functors, these yield the commutative shifted symplectic and Poisson structures, unifying various approaches and demonstrating the robust foundational character of non-degenerate shifted Poisson geometry in both commutative and non-commutative contexts.

Key functoriality includes compatibility with the commutative shadow ($A \mapsto A/[A,A]$), with induced non-degenerate shifted Poisson structures, and with the structure of Calabi–Yau dg-categories via cyclic (co)homology classes.

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Non-degenerate shifted Poisson structures thus serve as the rigorous algebraic and derived-geometric incarnation of higher Poisson geometry. They encode, via the Maurer–Cartan formalism and non-degeneracy, the simultaneous presence of both rich infinitesimal symplectic geometry and higher categorical and representation-theoretic data in derived moduli and non-commutative geometry. The theory is central to the deformation quantization of moduli spaces, the study of symplectic and Poisson moduli, representation theory of quantum groups, and modern approaches to vanishing cycles and microlocal sheaf theory.