If ais a symmetric frobenius algebra, the linear form will be denoted. It is shown that every frobenius algebra over a commutative ring determines a class of solutions of the quantum yangbaxter equation, which forms a subbimodule of its tensor square. Some applications of frobenius algebras to hopf algebras martin lorenz. There is also a different, mostly unrelated notion of the symmetric algebra of a vector space. Computing the index of lie algebras estonian academy publishers. On the other hand there is another infinitedimensional, in physical setting frobenius algebra sitting inside the classical bv fields. The frobenius semiradical of a lie algebra sciencedirect. There exists a algebra automorphism of such that for all. It is that ambient tensor category which knows if the frobenius algebra describes a topological or a conformal field theory in 2d. Frobenius algebras and 2d topological quantum eld theories. In addition, an equivalent relation between solutions of the classical homyangbaxter equation chybe and quasi frobenius hom lie algebras is also discussed. Orbifold frobenius algebras, cobordisms and monodromies 7 gmodules on the cohomology and the milnor ring w. The frobenius algebras relevant for cft dont live in vect, but in some other modular tensor category, usually that of representations of some chiral vertex operator algebra. Proof let and define by for all clearly and so by this lemma, there exists a unique such that for all so for all define the map by for all so we need to prove that is a algebra automorphism.
The finite dimensional 0 algebra r with identity is a frobenius algebra if and only if there is a linear form f e uom0ir, 0, the kernel of which contains no left or right ideal different from 0. We introduce pseudounimodular poisson algebras by generalizing unimodular poisson algebras, and investigate batalinvilkovisky structures on their cohomology algebras. In particular, a form of frobenius algebras has been used to give an algebraic axiomatization of or thonormal bases and observables cp07, cpv09. A representation of an associative algebra aalso called a left. Geometric methods in problems of analysis and algebra in russian, yaroslavsk state univ.
The frobenius semirad ica l of a lie algebra article pdf available in journal of algebra 2731. A lie algebra 9 over an arbitrary field is a frobenius lie algebra if there is a linear form l is an element of g whose stabilizer with respect to the coadjoint representation of 0, i. Modules, comodules and cotensor products over frobenius. Meanders and frobenius seaweed lie algebras coll, vincent, giaquinto, anthony, and magnant, colton, journal of generalized lie theory and applications, 2011. The essence of the latter proposition goes back to d. Koszul algebras and the frobenius automorphism 349 by the noetherskolem theorem, l l for all simple smodules l if. Berkson1 let c be a field of characteristic po, l a restricted finite dimensional lie algebra over c and ml its restricted enveloping algebra or m algebra 3, p. We show that the isomorphism class of a twostep solvable lie poset subalgebra of a semisimple lie algebra is determined by its dimension.
The only prerequisite for this volume is a basic knowledge of linear algebra and some results of the first volume. Then we show how to explicitly construct a coalgebra given a nitedimensional algebra. Frobenius algebra in segals story encodes the nonperturbative quantum result. Let c be a field of characteristic po, l a restricted finite dimen sional lie algebra over c and ml its restricted enveloping algebra or. Classification of frobenius lie algebras of dimension. We examine this graph in several ways with a goal of determining families of frobenius index zero seaweed. It gives a mirror partner for the theory of genus 0 gromovwitten invariants. The index of a seaweed lie algebra can be computed from its associated meander graph. Frobenius algebras and 2d qft the string coffee table. This book, written for undergraduate math students, describes a striking connection between topology and algebra, expressed by the theorem that 2d topological quantum field theories are the same as commutative frobenius algebras. A lie algebra g is called frobenius if there is a linear functional.
In physical terms, these two structures are related by spectral flow. Pdf the binary spectrum of typea frobenius lie poset. We construct a generalization of the variations of hodge structures on calabiyau manifolds. Introduction algebras, coalgebras, and frobenius algebras. If,, is a quasi frobenius lie algebra, one can define on another bilinear product by the formula. Subsequently, gerstenhaber and giaquinto asserted, in particular 12. We investigate the properties of principal elements of frobenius lie algebras, following the work of m. Some applications of frobenius algebras to hopf algebras. Co frobenius coalgebras were rst introduced by lin in l as a dualization of frobenius.
For any frobenius poisson algebra, all batalinvilkovisky operators on its poisson cochain complex are described. The principal element of a frobenius lie algebra core. This chapter starts with a reformulation and proof of the classical result of frobenius that characterizes the finite dimensional algebras over a field for which the left and right regular representations are equivalent, giving the brauernesbittnakayama criteria for a finite dimensional algebra to be a frobenius algebra or a symmetric one. The purpose of this paper is two fold, each of which is deeply related to k. Pdf classification of frobenius, twostep solvable lie. Our analysis gives two new families of frobenius seaweed algebras as well as elementary proofs of known families of such lie algebras. Such a result is derived from a discussion on the structure of 2step solvable frobenius lie algebras and a complete characterization of their associated left symmetric algebra lsa structure. Let r be a frobenius algebra and let be a bilinear form on r guaranteed by theorem 1. Pdf on properties of principal elements of frobenius lie. Homyangbaxter equations and frobenius monoidal homalgebras.
This paper is devoted to study frobenius poisson algebras. We show that the resulting structure is closely related to a frobenius algebra. Frobenius algebras and 2d topological quantum field theories. Moreover, the shapes of connected components of the stable auslanderreiten quivers of frobenius algebras are described. He classified all the algebraic frobenius algebras up to dimension 6. Frobenius algebras and coalgebras rolf farnsteiner throughout, we shall be working over a. The structure of frobenius algebras and separable algebras. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The objective of this lecture is to characterize nondegenerate associative forms on. The frobenius semiradical of a lie algebra alfons i. In this chapter we discuss elementary properties of lie groups, lie algebras. The following lemma is abrams characterization of frobenius algebras in terms of comultiplication.
A finite dimensional lie algebra f is frobenius if there is a linear frobenius functional f. On properties of principal elements of frobenius lie. For our purposes, the new characterization of frobenius algebras presented below is very useful. Modules, comodules, and cotensor products over frobenius. In 21 the concept of frobenius manifold is introduced and discussed. We prove that any lie algebra with a left symmetric algebra structure can be embedded, in a natural way, as a. So there exists a bilinear form which is nondegenerate and for all. The principal element of f is then the unique element. To be even more precise, one structure carries the natural. Equivalence with pre lie algebras with nondegenerate invariant skewsymmetric bilinear form. Given a lie algebra g with subalgebra h, if v is a representation of h, the induced representation of g is ug uh v this satis es the frobenius reciprocity relation.
A lie algebra lover k is called frobenius if lis a nitely. We examine this graph in several ways with a goal of determining families of frobenius index zero seaweed algebras. Frobenius algebras an algebra a over a field k is called frobenius when it is finite dimensional and equipped with a linear function e. Any matrix algebra defined over a field k is a frobenius algebra with frobenius form. Frobenius manifolds and formality of lie algebras of. We further establish that all such algebras are absolutely rigid. Orbifold frobenius algebras, cobordisms and monodromies. We examine this graph in several ways with a goal of determining families of frobenius. Via frobeniuscoordinates, kadisongavea simpli ed proof see5. For completeness and later use, we include a proof. In the present paper, we give a sufficient condition for this property to be satisfied, for any number of commuting matrices, for any arbitrary field k. Semisimple frobenius manifolds and conformal algebras.
A hom lie algebra in is a triple consisting of vector space, bilinear map called the bracket, and. Zhu on the degrees of irreducible representations, the socalled class equation. Request pdf classification of frobenius lie algebras of dimension. Constructing the extended haagerup planar algebra bigelow, stephen, peters, emily, morrison, scott, and snyder, noah, acta mathematica, 2012. With every finite dimensional restricted lie algebra there is asso ciated a family of finite dimensional associative frobenius algebras, the socalled reduced.