arXiv subjects

Serkan Sütlü

Publications and source records attributed to Serkan Sütlü.

At least 19 recordsLinked to original sources

arXiv2026

Couplings of $3$-anchored Bundles

This work develops an algebraic framework for merging two $3$-anchored bundles over the same base manifold, equipped with mutual actions and two twisted cocycle terms, so as to obtain a $3$-Lie algebroid structure on the corresponding Whitney sum. We also record the purely algebraic counterpart of this construction, namely the bicocycle double cross product $3$-Lie algebra, obtained by removing the anchor and Leibniz-type compatibility conditions. The resulting framework provides a unified setting for $3$-Lie algebroids and contains, as special cases, unified products, double cross products, semi-direct products, cocycle extensions, and direct products.

math.DG↗

arXiv2026

Constructions of $3$-Lie algebroids

The paper investigates the construction of Lie algebroids and $3$-Lie algebroids via connections generated by finite families of differential operators and dual sections. We first recall the description of Lie and $n$-Lie algebroid brackets in terms of connections, and introduce an $n$-curvature operator whose $n$-Bianchi identity characterizes the fundamental identity. We then provide sufficient conditions under which such generating families determine Lie algebroid and $3$-Lie algebroid structures. The construction extends the single-operator approach and covers natural examples such as the Jacobi Lie algebroid. As an application, we construct a concrete $3$-Lie algebroid structure arising from Poisson Lie algebroid data.

math.DG↗

arXiv2025

On Product Lie Algebroids, and Collective Motion

This work explores the geometrical/algebraic framework of Lie algebroids, with a specific focus on the decoupling and coupling phenomena within the bicocycle double cross product realization. The bicocycle double cross product theory serves as the most general method for (de)coupling an algebroid into the direct sum of two vector bundles in the presence of mutual \textit{representations}, along with two twisted cocycle terms. Consequently, it encompasses unified product, double cross product (matched pairs), semi-direct product, and cocycle extension frameworks as particular instances. In addition to algebraic constructions, the research extends to both reversible and irreversible Lagrangian and Hamiltonian dynamics on (de)coupled Lie algebroids, as well as Euler-Poincaré-(Herglotz) and Lie-Poisson-(Herglotz) dynamics on (de)coupled Lie algebras, providing insights into potential physical applications.

math.DG↗

arXiv2024

Conformal and Contact Kinetic Dynamics and Their Geometrization

We propose a conformal generalization of the reversible Vlasov equation of kinetic plasma dynamics, called conformal kinetic theory. In order to arrive at this formalism, we start with the conformal Hamiltonian dynamics of particles and lift it to the dynamical formulation of the associated kinetic theory. The resulting theory represents a simple example of a geometric pathway from dissipative particle motion to dissipative kinetic motion. We also derive the kinetic equations of a continuum of particles governed by the contact Hamiltonian dynamics, which may be interpreted in the context of relativistic mechanics. Once again we start with the contact Hamiltonian dynamics and lift it to a kinetic theory, called contact kinetic dynamics. Finally, we project the contact kinetic theory to conformal kinetic theory so that they form a geometric hierarchy.

math-ph↗

arXiv2024

Quantum van Est Isomorphism

Motivated by the fact that the Hopf-cyclic (co)homologies of function algebras over Lie groups and universal enveloping algebras over Lie algebras capture the Lie group and Lie algebra (co)homologies, we hereby upgrade the classical van Est isomorphism to ones between the Hopf-cyclic (co)homologies of quantized algebras of functions and quantized universal enveloping algebras, both in $h$-adic and $q$-deformation frameworks.

math.QA↗

arXiv2024

Dynamics Over Homogeneous Spaces

We present the Euler-Lagrange and Hamilton's equations for a system whose configuration space is a unified product Lie group $G=M\bowtie_γ H$, for some $γ:M\times M \to H$. By reduction, then, we obtain the Euler-Lagrange type and Hamilton's type equations of the same form for the quotient space $M\cong G/H$, although it is not necessarily a Lie group. We observe, through further reduction, that it is possible to formulate the Euler-Poincaré type and Lie-Poisson type equations on the corresponding quotient $\mathfrak{m}\cong \mathfrak{g}/\mathfrak{h}$ of Lie algebras, which is not a priori a Lie algebra. Moreover, we realize the $n$th order iterated tangent group $T^{(n)}G$ of a Lie group $G$ as an extension of the $n$th order tangent group $T^nG$ of the same type. More precisely, $\mathfrak{g}$ being the Lie algebra of $G$, $T^{(n)}G \cong \mathfrak{g}^{\times \,2^n-1-n} \bowtie_γT^nG$ for some $γ:\mathfrak{g}^{\times \,2^n-1-n} \times \mathfrak{g}^{\times \,2^n-1-n} \to T^nG$. We thus obtain the $n$th order Euler-Lagrange (and then the $n$th order Euler-Poincaré) equations over $T^nG$ by reduction from those on $T(T^{n-1}G)$. Finally, we illustrate our results in the realm of the Kepler problem, and the non-linear tokamak plasma dynamics.

math.DG↗

arXiv2022

Tulczyjew's Triplet with an Ehresmann connection I: Trivialization and Reduction

We study the trivialization and the reduction of the Tulczyjew's triplet, in the presence of a symmetry and an Ehresmann connection associated to it. We thus obtain trivializations and reductions of iterated tangent and cotangent bundles $T^*TQ$, $TT^*Q$ and $T^*T^*Q$. Accordingly, the symplectomorphisms between these manifolds are properly trivialized and reduced.

math-ph↗

arXiv2021

Tulczyjew's Triplet for Lie Groups III : Higher Order Dynamics and Reductions for Iterated Bundles

Given a Lie group $G$, we elaborate the dynamics on $T^*T^*G$ and $T^*TG$, which is given by a Hamiltonian, as well as the dynamics on the Tulczyjew symplectic space $TT^*G$, which may be defined by a Lagrangian or a Hamiltonian function. As the trivializations we adapted respect the group structures of the iterated bundles, we exploit all possible subgroup reductions (Poisson, symplectic or both) of higher order dynamics.

math.SG↗

arXiv2021

Matched pair analysis of the Vlasov plasma

We present the Hamiltonian (Lie-Poisson) analysis of the Vlasov plasma, and the dynamics of its kinetic moments, from the matched pair decomposition point of view. We express these (Lie-Poisson) systems as couplings of \textit{mutually interacting} (Lie-Poisson) subdynamics. The mutual interaction is beyond the well-known semi-direct product theory. Accordingly, as the geometric framework of the present discussion, we address the \textit{matched pair Lie-Poisson} formulation allowing mutual interactions. Moreover, both for the kinetic moments and the Vlasov plasma cases, we observe that one of the constitutive subdynamics is the compressible isentropic fluid flow, and the other is the dynamics of the kinetic moments of order $\geq 2$. In this regard, the algebraic/geometric (matched pair) decomposition that we offer, is in perfect harmony with the physical intuition. To complete the discussion, we present a momentum formulation of the Vlasov plasma, along with its matched pair decomposition.

math-ph↗

arXiv2021

Inertial Hopf-cyclic homology

We construct, study, and apply a characteristic map from the relative periodic cyclic homology of the quotient map for a group action to the periodic Hopf-cyclic homology with coefficients associated with inertia of the action. This result admits, and in fact, comes from, its noncommutative-geometric, or quantized, counterpart. The crucial ingredient is the construction of the appropriate quantization of the cyclic nerve of the action groupoid, the cyclic object related to inertia, as the Connes-cyclic dual of a Hopf-cyclic object with coefficients in some stable anti-Yetter--Drinfeld module quantizing the Brylinski space. For the Hopf-Galois quantization of the case of trivial inertia, we find a non-trivial identification of our characteristic map with a well-known isomorphism of Jara--Ştefan. In presence of nontrivial inertia, for an analytic ramified Galois double cover, we show that the cokernel of our characteristic map, which can be interpreted as an invariant of inertia modulo topology of the maximal free action, is supported on the branch locus of the quotient map. Finally, we use our inertial Hopf-cyclic object to construct a new invariant of finite-dimensional algebras.

math.KT↗

arXiv2021

On Extensions, Lie-Poisson Systems, and Dissipations

On the dual space of \textit{extended structure}, equations governing the collective motion of two mutually interacting Lie-Poisson systems are derived. By including a twisted 2-cocycle term, this novel construction is providing the most general realization of (de)coupling of Lie-Poisson systems. A double cross sum (matched pair) of 2-cocycle extensions are constructed. The conditions are determined for this double cross sum to be a 2-cocycle extension by itself. On the dual spaces, Lie-Poisson equations are computed. We complement the discussion by presenting a double cross sum of some symmetric brackets, such as double bracket, Cartan-Killing bracket, Casimir dissipation bracket, and Hamilton dissipation bracket. Accordingly, the collective motion of two mutually interacting irreversible dynamics, as well as mutually interacting metriplectic flows, are obtained. The theoretical results are illustrated in three examples. As an infinite-dimensional physical model, decompositions of the BBGKY hierarchy are presented. As finite-dimensional examples, the coupling of two Heisenberg algebras and coupling of two copies of $3D$ dynamics are studied.

math-ph↗

arXiv2020

Second Order Lagrangian Dynamics On Double Cross Product Groups

We observe that the iterated tangent group of a Lie group may be realized as a double cross product of the 2nd order tangent group, with the Lie algebra of the base Lie group. Based on this observation, we derive the 2nd order Euler-Lagrange equations on the 2nd order tangent group from the 1st order Euler-Lagrange equations on the iterated tangent group. We also present in detail the 2nd order Lagrangian dynamics on the 2nd order tangent group of a double cross product group.

math-ph↗

arXiv2019

Homology of quantum linear groups

For every $n\geq 1$, we calculate the Hochschild homology of the quantum monoids $M_q(n)$, and the quantum groups $GL_q(n)$ and $SL_q(n)$ with coefficients in a 1-dimensional module coming from a modular pair in involution.

math.KT↗

arXiv2019

Matched pairs of discrete dynamical systems

Matched pairs of Lie groupoids and Lie algebroids are studied. Discrete Euler-Lagrange equations are written for the matched pairs of Lie groupoids. As such, a geometric framework to analyse a discrete system by decomposing it into two mutually interacting subsystems is established. Two examples are provided to illustrate this strategy; the discrete dynamics on the trivial groupoid, and the discrete dynamics on the special linear group.

math-ph↗

arXiv2018

On the Hochschild homology of smash biproducts

We develop a new spectral sequence in order to calculate Hochschild homology of smash biproducts (also called twisted tensor products) of unital associative algebras $A\# B$ provided one of $A$ or $B$ has Hochschild dimension less than 2. We use this spectral sequence to calculate Hochschild homology of quantum tori, multiparametric quantum affine spaces, quantum complete intersections, quantum Weyl algebras, deformed completed Weyl algebras, and finally the algebra $M_q(2)$ of quantum $2\times 2$-matrices.

math.KT↗

arXiv2018

Topological Hopf algebras and their Hopf-cyclic cohomology

A natural extension of the Hopf-cyclic cohomology, with coefficients, is introduced to encompass topological Hopf algebras. The topological theory allows to work with infinite dimensional Lie algebras. Furthermore, the category of coefficients (AYD modules) over a topological Lie algebra and those over its universal enveloping (Hopf) algebra are isomorphic. For topological Hopf algebras, the category of coefficients is identified with the representation category of a topological algebra called the anti-Drinfeld double. Finally, a topological van Est type isomorphism is detailed, connecting the Hopf-cyclic cohomology to the relative Lie algebra cohomology with respect to a maximal compact subalgebra.

math.KT↗

arXiv2017

The asymptotic Connes-Moscovici characteristic map and the index cocycles

We show that the (even and odd) index cocycles for theta-summable Fredholm modules are in the image of the Connes-Moscovici characteristic map. To show this, we first define a new range of asymptotic cohomologies, and then we extend the Connes-Moscovici characteristic map to our setting. The ordinary periodic cyclic cohomology and the entire cyclic cohomology appear as two instances of this setup. We then construct an asymptotic characteristic class, defined independently from the underlying Fredholm module. Paired with the $K$-theory, the image of this class under the characteristic map yields a non-zero scalar multiple of the index in the even case, and the spectral flow in the odd case.

math.KT↗

arXiv2016

Lagrangian Dynamics on Matched Pairs

Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the Euler-Lagrange equations on the trivialized matched pair of tangent groups, as well as the Euler-Poincaré equations on the matched pair of Lie algebras. We show explicitly how these equations cover those of the semi-direct product theory. In particular, we study the trivialized, and the reduced Lagrangian dynamics on the group $SL(2,\mathbb{C})$.

math-ph↗