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Cayley Theorems for Loday Algebras

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Abstract

Loday’s notoriously elusive “coquecigrues” are meant to relate to Leibniz algebras in the same various ways that groups relate to Lie algebras. However, with the current approaches based on digroups, deadlock has been reached at the analogues of Lie’s Third Theorem. Here, adjoint representations appear in the places where regular representations should be expected. The present work, intended as a stimulus to new approaches to the problem, proposes more symmetrical versions of the algebras involved. The fundamental guiding principle is to maintain both left and right actions on a completely equal footing. A coherent and cumulative series of Cayley theorems gives concrete representations of abstract split versions of semigroups, monoids, and groups, based upon the Galois theory of “symmetries of symmetries”. Interpreted within monoidal categories, the new group-like objects we present provide a complete left/right split of Hopf algebra structure. The Cayley embedding appears intrinsically as the left/right symmetric part of the coassociativity diagram.

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Notes

  1. While this diagrammatic specification of the split Hopf algebra structure takes up more space than compact syntax with Heynemann-Sweedler notation (as in [36, (4.1)–(4.4)], for example), it is much more transparent, particularly where geometric symmetry of the diagrams reflects the logical symmetry and duality of the theory.

  2. Turnstiles \(\dashv ,\vdash \) have previously been the notation of choice for the left- and right-handed products \(\lhd ,\rhd \) as they appear in (1.3). However, since a turnstile bars access with its horizontal part, the bar unit e in (1.3) would confusingly appear on the side away from the bar of the turnstile. The triangular product symbols of (1.3), which will be used throughout this paper, represent left- and right-handed versions of the multiplication \(\nabla \) in a Hopf algebra. Turnstiles will be used for the left- and right-handed convolution products of Sect. 6.7.

  3. The left and right multiplication notations used in the specifications of \(\alpha \) and \(\beta \) follow (2.3) and (2.4).

  4. In the context of conformal algebras (compare [38, Ex. 2.7]), a referee notes Kolesnikov’s terminology of 0-identities for the bar side irrelevance identites [13].

  5. Specifically, discarding the formal specification of E and the inversion operations (cf. [40, p. 287]).

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Acknowledgements

The author is grateful to the referee for their many detailed and insightful comments on an earlier version of the manuscript.

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  1. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Jonathan D. H. Smith

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Smith, J.D.H. Cayley Theorems for Loday Algebras. Results Math 77, 218 (2022). https://doi.org/10.1007/s00025-022-01748-8

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