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Mikołaj Rotkiewicz

Publications and source records attributed to Mikołaj Rotkiewicz.

6 recordsLinked to original sources
arXiv2026

Equivalence functors in graded supergeometry

It has recently been proved that the category of N-manifolds of degree $n$, that is, $\mathbb N$-graded supermanifolds of degree $n$ for which the parity agrees with the gradation, is equivalent to the category of purely even $n$-tuple vector superbundles equipped with a suitable action of the symmetric group $S_n$ permuting the vector bundle structures. This equivalence may be interpreted as a `desuperization' of N-manifolds. In the present paper, we place this result within a broader framework of graded structures on supermanifolds and explicitly describe several canonical equivalences between the corresponding categories in a purely geometric, constructive manner. The desuperization equivalence functor appears as a composition of some of these canonical equivalences. Our constructions are entirely canonical and rely on standard tools of supergeometry, including iterated tangent functors, parity reversion in vector superbundles, and the interpretation of $n$-tuple vector bundles in terms of commuting Euler vector fields associated with the underlying vector bundle structures.

math.DG↗
arXiv2024

Exploring the Structure of Higher Algebroids

The notion of a \emph{higher-order algebroid}, as introduced by Jóźwikowski and Rotkiewicz in their work \emph{Higher-order analogs of Lie algebroids via vector bundle comorphisms} (SIGMA, 2018), generalizes the concepts of a higher-order tangent bundle $τ^k_M: \mathrm{T}^k M \to M$ and a (Lie) algebroid. This idea is based on a (vector bundle) comorphism approach to (Lie) algebroids and the reduction procedure of homotopies from the level of Lie groupoids to that of Lie algebroids. In brief, an alternative description of a Lie algebroid $(A, [\cdot, \cdot], \sharp)$ is a vector bundle comorphism $κ$, defined as the dual of the Poisson map $\varepsilon: \mathrm{T}^\ast A \to \mathrm{T} A^\ast$ associated with the Lie algebroid $A$. The framework of comorphisms has proven to be a suitable language for describing higher-order analogues of Lie algebroids from the perspective of the role played by (Lie) algebroids in geometric mechanics. In this work, we uncover the classical algebraic structures underlying the somewhat mysterious description of higher-order algebroids through comorphisms. For the case $k=2$, we establish a one-to-one correspondence between higher-order Lie algebroids and pairs consisting of a two-term representation (up to homotopy) of a Lie algebroid and a morphism to the adjoint representation of this algebroid.

math.DG↗
arXiv2018

Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms

We introduce the concept of a higher algebroid, generalizing the notions of an algebroid and a higher tangent bundle. Our ideas are based on a description of (Lie) algebroids as vector bundle comorphisms - differential relations of a special kind. In our approach higher algebroids are vector bundle comorphism between graded-linear bundles satisfying natural axioms. We provide natural examples and discuss applications in geometric mechanics.

math.DG↗
arXiv2016

Duality for graded manifolds

We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different category than the initial ones, namely graded polynomial (co)algebra bundles and free graded Weil (co)algebra bundles. Our results are then applied to obtain elegant characterizations of double vector bundles and graded bundles of degree 2. All these results have their supergeometric counterparts. For instance, we give a simple proof of a nice characterisation of $N$-manifolds of degree 2, announced in the literature.

math.DG↗
arXiv2014

Prototypes of higher algebroids with applications to variational calculus

Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a differential relation of special kind (a Zakrzewski morphism). In the paper we investigate basic properties of higher algebroids and show some applications. Namely, we develop a geometric framework for variational calculus on higher algebroids (including both forces and momenta). Such a formalism covers simultaneously variational problems on higher tangent bundles, first-order problems on algebroids and higher-order problems reduced by symmetries.

math.DG↗