arXiv subjects

Geoffrey Powell

Publications and source records attributed to Geoffrey Powell.

At least 19 recordsLinked to original sources

arXiv2026

Relating Brauer categories, Koszul complexes, and graph complexes

The purpose of this paper is to investigate the relationship between hairy graph complexes associated to cyclic operads and their counterparts for operads (and, more generally, dioperads). This is based on the author's interpretation of these as Koszul complexes for the associated modules over the respective appropriate twisted downward (walled) Brauer category. The general question of relating such Koszul complexes is addressed by analysing the relationships between the respective twisted Brauer-type categories, proceeding through a direct analysis. The passage from the walled to unwalled context involves functors induced by the disjoint union of finite sets. As an application, for the cyclic operad associated to an operad, this leads to an explicit relation between the respective (hairy) graph homologies.

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arXiv2025

Cyclic operads, Koszul complexes, and hairy graph complexes

In this paper, we revisit the construction of the hairy graph complexes associated to a cyclic operad, by exploiting modules over the appropriate twisted linearization of the downward Brauer category (and working over a field of characteristic zero). The different flavours (even or odd) of complexes appear as forms of Koszul complexes; the Koszul property of the linear category provides an elegant homological interpretation of their homology. This approach allows a second form of Koszul complex to enter the picture. For the `even' flavour, this corresponds to a precursor of the Chevalley-Eilenberg complex of the Conant-Vogtmann Lie algebra associated to a cyclic operad and a symplectic vector space (generalizing Kontsevich's Lie algebras). Again, the cohomology of the Koszul complex has an elegant interpretation. This sheds light on the relationship between the unstable case and Kontsevich's identification (generalized by Conant and Vogtmann) of the homology in the infinite-dimensional case with a form of graph homology (in the even case). We observe that this is already of interest in the case of an algebra with involution, viewed as a cyclic operad.

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arXiv2025

Operads, modules over walled Brauer categories, and Koszul complexes

We investigate certain complexes that are associated to an operad $\mathscr{O}$ in $k$-vector spaces, where $k$ is a field of characteristic $0$. This exploits the study of modules over the $k$-linearization of the upward walled Brauer category, $k\mathsf{uwb}$ (respectively of the downward walled Brauer category, $k\mathsf{dwb}$). These are Koszul over $k(\mathbf{FB \times FB})$, where $\mathbf{FB}$ is the category of finite sets and bijections. We show that the Chevalley-Eilenberg complex for the Lie algebra of derivations $\mathrm{Der} (\mathscr{O} (V))$ of the free $\mathscr{O}$-algebra on a finite-dimensional vector space $V$ has a precursor given by the Koszul complex on an explicit module over $(k\mathsf{dwb})_-$ (a twisted $k$-linearization of $\mathsf{dwb}$); this module is constructed naturally from the operad $\mathscr{O}$. Following Dotsenko, we also consider the more general case where a {\em wheeled} term is included. This identification exploits functoriality with respect to the category of finite-dimensional $k$-vector spaces with morphisms taken to be split monomorphisms, together with the relationship with functors on the upward (and downward) walled Brauer category. We also exploit methods developed by Sam and Snowden for investigating the stabilization of the families of representations of the general linear groups associated to a functor on the category of split monomorphisms between $k$-vector spaces. We give a new perspective on the results of Dotsenko, who investigated the stable homology of Lie algebras of derivations and established a link with the wheeled bar construction for wheeled operads. In particular, we explain why one of the Koszul complexes that we consider should be considered as the appropriate form of hairy graph complex for operads, by analogy with the case of cyclic operads.

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arXiv2025

Lie algebra homology with coefficients tensor products of the adjoint representation in relative polynomial degree 2

The homology of free Lie algebras with coefficients in tensor products of the adjoint representation working over Q contains important information on the homological properties of polynomial outer functors on free groups. The latter category was introduced in joint work with Vespa, motivated by the study of higher Hochschild homology of wedges of circles. There is a splitting of this homology by polynomial degree (for polynomiality with respect to the generators of the free Lie algebra) and one can consider the polynomial degree relative to the number of tensor factors in the coefficients. It suffices to consider the Lie algebra homology in homological degree one; this vanishes in relative degree 0 and is readily calculated in relative degree 1. This paper calculates the homology in relative degree 2, which presents interesting features. This confirms a conjecture of Gadish and Hainaut.

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arXiv2024

Functors on the category of finite sets revisited

We study the structure of the category of representations of $\mathbf{FA}$, the category of finite sets and all maps, mostly working over a field of characteristic zero. This category is not semi-simple and exhibits interesting features. We first construct the simple representations, recovering the classification given by Wiltshire-Gordon. The construction given here also yields explicit descriptions of the indecomposable projectives. These results are used to give a convenient set of projective generators of the category of representations of $\mathbf{FA}$ and hence a Morita equivalence result. This is used to explain how to calculate the multiplicities of the composition factors of an arbitrary object, based only on its underlying $\mathbf{FB}$-representation, where $\mathbf{FB}$ is the category of finite sets and bijections. This is applied to show how to calculate the morphism spaces between projectives in our chosen set of generators, as well as for a closely related family of objects (the significance of which can be shown by relative nonhomogeneous Koszul duality theory).

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arXiv2024

Filtering the linearization of the category of surjections

A filtration of the morphisms of the $k$-linearization $k \mathbf{FS}$ of the category $\mathbf{FS}$ of finite sets and surjections is constructed using a natural $k \mathbf{FI}^{op}$-module structure induced by restriction, where $\mathbf{FI}$ is the category of finite sets and injections. In particular, this yields the `primitive' subcategory $ k \mathbf{FS}^0 \subset k \mathbf{FS}$ that is of independent interest; for example, the category of $k \mathbf{FS}^0$-modules is closely related to the category of $k \mathbf{FA}$-modules, where $\mathbf{FA}$ is the category of finite sets and all maps. Working over a field of characteristic zero, the subquotients of this filtration are identified as bimodules over $k \mathbf{FB}$, where $\mathbf{FB}$ is the category of finite sets and bijections, also exhibiting and exploiting additional structure. In particular, this describes the underlying $k \mathbf{FB}$-bimodule of $k \mathbf{FS}^0$.

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arXiv2024

Relative nonhomogeneous Koszul duality for PROPs associated to nonaugmented operads

The purpose of this paper is to show how Positselski's relative nonhomogeneous Koszul duality theory applies when studying the linear category underlying the PROP associated to a (non-augmented) operad of a certain form, in particular assuming that the reduced part of the operad is binary quadratic. In this case, the linear category has both a left augmentation and a right augmentation (corresponding to different units), using Positselski's terminology. The general theory provides two associated linear differential graded (DG) categories; indeed, in this framework, one can work entirely within the DG realm, as opposed to the curved setting required for Positselski's general theory. Moreover, DG modules over DG categories are related by adjunctions. When the reduced part of the operad is Koszul (working over a field of characteristic zero), the relative Koszul duality theory shows that there is a Koszul-type equivalence between the appropriate homotopy categories of DG modules. This gives a form of Koszul duality relationship between the above DG categories. This is illustrated by the case of the operad encoding unital, commutative associative algebras, extending the classical Koszul duality between commutative associative algebras and Lie algebras. In this case, the associated linear category is the linearization of the category of finite sets and all maps. The relative nonhomogeneous Koszul duality theory relates its derived category to the respective homotopy categories of modules over two explicit linear DG categories.

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arXiv2023

On the Passi and the Mal'cev functors

The author has shown that the category of analytic contravariant functors on $\mathbf{gr}$, the category of finitely-generated free groups, is equivalent to the category of left modules over the PROP associated to the Lie operad, working over $\mathbb{Q}$. This exploited properties of the polynomial filtration of the category of contravariant functors on $\mathbf{gr}$. The first purpose of this paper is to strengthen the corresponding result for covariant functors on $\mathbf{gr}$. This involves introducing the appropriate analogue of the category of analytic contravariant functors, namely a certain category of towers of polynomial functors on $\mathbf{gr}$. This category is abelian and has a natural symmetric monoidal structure induced by the usual tensor product of functors. Moreover, the projective generators of this category are described in terms of the Mal'cev functors that are introduced here. It follows that this category is equivalent to the category of right modules over the PROP associated to the Lie operad. As a fundamental example, the Passi functors arising from the group ring functors are described explicitly. The theory is applied to consider bifunctors on $\mathbf{gr}$. This allows the $\mathbb{Q}$-linearization of the category of free groups to be described, up to polynomial filtration. As a stronger application of the theory, this is generalized to the Casimir PROP associated to the Lie operad, as studied by Hinich and Vaintrob. Up to polynomial filtration, this recovers the category $\mathbf{A}$ introduced by Habiro and Massuyeau in their study of bottom tangles in handlebodies.

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arXiv2023

On fundamental structure underlying Lie algebra homology with coefficients tensor products of the adjoint representation

This paper exhibits fundamental structure underlying Lie algebra homology with coefficients in tensor products of the adjoint representation, mostly focusing upon the case of free Lie algebras. The main result yields a DG category that is constructed from the PROP associated to the Lie operad. Underlying this is a two-term complex of bimodules over this PROP; it is a quotient of the universal Chevalley-Eilenberg complex. The homology of this DG category is intimately related to outer functors over free groups (introduced in earlier joint work with Vespa). This uses the author's previous results relating functors on free groups to representations of the PROP associated to the Lie operad. This gives a direct algebraic explanation as to why the degree one homology should correspond to an outer functor. Hitherto, the only known argument relied upon the relationship with the higher Hochschild homology functors that arise from the work of Turchin and Willwacher.

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arXiv2022

On the FI-homology of the injective cogenerators

The purpose of this paper is to give information on the FI-homology of the standard injective cogenerators of the category of FI-modules, where FI is the category of finite sets and injections. Working over a field k of characteristic zero, a full calculation is given in homological degree zero and a conjectural description in higher homological degree. The proof of the main theorem reduces to a calculation in representation theory of the symmetric groups, exploiting the Young orthonormal basis.

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arXiv2022

Baby bead representations

This paper is motivated by the study of Turchin and Willwacher's bead representations. The problem is reformulated here in terms of the Lie algebra homology of a free Lie algebra with coefficients in tensor products of the adjoint representation. The main idea is to exploit the truncation of the coefficients given by killing Lie brackets of length greater than two. Although this truncation is brutal, it retains significant and highly non-trivial information, as exhibited by explicit results. A d\'evissage is used that splits the problem into two steps, separating out a `homology' calculation from `antisymmetrization'. This involves some auxiliary categories, including a generalization of the upper walled Brauer category. This approach passes through the `baby bead representations' of the title, for which complete results are obtained. As an application, the composition factors of Turchin and Willwacher's bead representations are calculated for a new infinite family.

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arXiv2022

Outer functors and a general operadic framework

For $\mathcal{O}$ an operad in $k$-vector spaces, the category $\mathcal{F}_\mathcal{O}$ is defined to be the category of $k$-linear functors from the PROP associated to $\mathcal{O}$ to $k$-vector spaces. Given $\mu \in \mathcal{O} (2)$ that satisfies a right Leibniz condition, the full subcategory $\mathcal{F}_\mathcal{O}^\mu \subset \mathcal{F}_\mathcal{O}$ is introduced here and its properties studied. This is motivated by the case of the Lie operad, where $\mu$ is taken to be the generator. By previous results of the author, when $k = \mathbb{Q}$, $\mathcal{F}_{Lie}$ is equivalent to the category of analytic functors on the opposite of the category $\mathbf{gr}$ of finitely-generated free groups. The main result shows that $\mathcal{F}_{Lie}^\mu$ identifies with the category of outer analytic functors, as introduced in earlier work of the author with Vespa. Using this identification, this theory has applications to the study of the higher Hochschild homology functors related to work of Turchin and Willwacher.

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arXiv2021

On analytic contravariant functors on free groups

Working over a field $k$ of characteristic zero, the category of analytic contravariant functors on the category of finitely-generated free groups is shown to be equivalent to the category of representations of the $k$-linear category associated to the Lie operad. Two proofs are given of this result. The first uses the original Ginzburg-Kapranov approach to Koszul duality of binary quadratic operads and the fact that the category of analytic contravariant functors is Koszul. The second proof proceeds by making the equivalence explicit using the $k$-linear category associated to the operad encoding unital associative algebras, which provides the `twisting bimodule'. A key ingredient is the Poincar\'e-Birkhoff-Witt theorem. Using the explicit formulation, it is shown how this equivalence reflects the tensor product on the category of analytic contravariant functors, relating this to the convolution product for representations of the category associated to the Lie operad.

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arXiv2021

The primitive filtration of the Leibniz complex

Pirashvili exhibited a small subcomplex of the Leibniz complex $(T(s \mathfrak{g}), d_{\mathrm{Leib}})$ of a Leibniz algebra $\mathfrak{g}$. The main result of this paper generalizes this result to show that the primitive filtration of $T(s\mathfrak{g})$ provides an increasing, exhaustive filtration of the Leibniz complex by subcomplexes, thus establishing a conjecture due to Loday. The associated spectral sequence is used to give a new proof of Pirashvili's conjecture that, when $\mathfrak{g}$ is a free Leibniz algebra, the homology of the Pirashvili complex is zero except in degree one. This result is then used to show that the desuspension of the Pirashvili complex carries a natural $L_\infty$-structure that induces the natural Lie algebra structure on the homology of the complex in degree zero.

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arXiv2021

Homological splitting results for modules over Leibniz algebras

A unified splitting result for Ext calculated in the category of modules over a Leibniz algebra is given for the case where coefficients are either both symmetric modules or both antisymmetric modules. This is a generalization of results of Loday and Pirashvili and others.

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arXiv2021

On derivations of free algebras over operads and the generalized divergence

For $\mathcal{O}$ a reduced operad, a generalized divergence from the derivations of a free $\mathcal{O}$-algebra to a suitable trace space is constructed. In the case of the Lie operad, this corresponds to Satoh's trace map and, for the associative operad, to the double divergence of Alekseev, Kawazumi, Kuno and Naef. The generalized divergence is shown to be a $1$-cocycle for the usual Lie algebra structure on derivations. These results place the previous constructions into a unified framework; moreover, they are natural with respect to the operad. An important new ingredient is the use of naturality with respect to the category of finite-rank free modules and split monomorphisms over a commutative ring $R$. This allows the notion of torsion for such functors to be exploited. Supposing that the ring $R$ is a PID and that the operad $\mathcal{O}$ is binary, the main result relates the kernel of the generalized divergence to the sub Lie algebra of the Lie algebra of derivations that is generated by the elements of degree one with respect to the grading induced by arity.

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arXiv2020

A Pirashvili-type theorem for functors on non-empty finite sets

Pirashvili's Dold-Kan type theorem for finite pointed sets follows from the identification in terms of surjections of the morphisms between the tensor powers of a functor playing the role of the augmentation ideal; these functors are projective. We give an unpointed analogue of this result: namely, we compute the morphisms between the tensor powers of the corresponding functor in the unpointed context. We also calculate the Ext groups between such objects, in particular showing that these functors are not projective; this is an important difference between the pointed and unpointed contexts. This work is motivated by our functorial analysis of the higher Hochschild homology of a wedge of circles.

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arXiv2018

Symmetric powers, Steenrod operations and representation stability

Working over the prime field F_p, the structure of the indecomposables Q^* for the action of the algebra of Steenrod reduced powers A(p) on the symmetric power functors S^* is studied by exploiting the theory of strict polynomial functors. In particular, working at the prime 2, representation stability is exhibited for certain related functors, leading to a conjectural representation stability description of quotients of Q^* arising from the polynomial filtration of symmetric powers.

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