IFFT-Equivariant Quantizations

F. Boniver; P. Mathonet

Home/Archive/arXiv

arXiv · math/0109032

IFFT-Equivariant Quantizations

F. Boniver·P. Mathonet

View original source ↗Read source document ↗Export metadata ↓

Abstract

The existence and uniqueness of quantizations that are equivariant with respect to conformal and projective Lie algebras of vector fields were recently obtained by Duval, Lecomte and Ovsienko. In order to do so, they computed spectra of some Casimir operators. We give an explicit formula for those spectra in the general framework of IFFT-algebras classified by Kobayashi and Nagano. We also define tree-like subsets of eigenspaces of those operators in which eigenvalues can be compared to show the existence of IFFT-equivariant quantizations. We apply our results to prove existence and uniqueness of quantizations that are equivariant with respect to the infinitesimal action of the symplectic (resp. pseudo-orhogonal) group on the corresponding Grassmann manifold of maximal isotropic subspaces.

Explore related subjects

math.RT math.QA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

F. Boniver, P. Mathonet. 2001-09-05. IFFT-Equivariant Quantizations. https://arxiv.org/abs/math/0109032

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A lecture note on covering theory in representation theory of algebras

Covering theory is an important tool in representation theory of algebras, however, the results and the proofs are scattered in the literature. We give an introduction to covering theory at a level as elementary as possible and in a more general way on some basic notions than the existing literature does.

math.RT↗

arXiv2026

On the Orthogonal Projections

Let $A$ be a finite-dimensional algebra and $e$ a rigid presentation. We give an explicit formula for the left adjoint to the inclusion $\operatorname{rep}(e^\perp)\hookrightarrow \operatorname{rep}(A)$ and construct a bijection between presentations in $\operatorname{rep}(e^\perp)$ and presentations compatible with $e$. The induced bijection on weights preserves generic $\operatorname{E}$-invariants and canonical decompositions. If $A$ is Jacobian, the reduction algebra is Jacobian for every rigid $e$. We prove mutation formulas for the signed completions and for the dimension vectors of the simple objects of $\operatorname{rep}(e^\perp)$, enabling an algorithm to find the projected quiver with potential when $e$ is extended-reachable. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.

math.RT↗

arXiv2026

Compatible gradings on dg models and cluster equivalence

Let $\La_1$ and $\La_2$ be finite-dimensional $\ta_2$-finite complex algebras of global dimension at most two. We prove that, if they are cluster equivalent, then they admit $\bbZ$-gradings for which the bounded derived categories of graded modules are triangle equivalent. The equivalence is not required to commute with the internal grading shifts. The proof uses common multigraded dg models, relative gradings on two-term silting objects, and covering quotients. We also consider tilting lifts of cluster-tilting objects with endomorphism algebras of global dimension at most two. In acyclic cluster type, the existence of such a lift for every cluster-tilting object implies piecewise heredity. We construct a counterexample without this assumption using a surface algebra associated with a one-holed torus.

math.RT↗