Lie algebras in symmetric monoidal categories

We study the algebras that are defined by identities in the symmetric monoidal categories; in particular, the Lie algebras. Some examples of these algebras appear in studying the knot invariants and the Rozansky-Witten invariants. The main result is the proof of the Westbury conjecture for a K3-surface: there exists a homomorphism from a universal simple Vogel algebra into a Lie algebra that describes the Rozansky-Witten invariants of a K3-surface. We construct a language that is necessary for discussing and solving this problem, and we formulate nine new problems.     

The Derived Category of Surface Algebras: the Case of the Torus with One Boundary Component

In this paper we refine the main result of a previous paper of the author with Grimeland on derived invariants of surface algebras. We restrict to the case where the surface is a torus with one boundary component and give an easily computable derived invariant for such surface algebras. This result permits to give answers to open questions on gentle algebras: it provides examples of gentle algebras with the same AG-invariant (in the sense of Avella-Alaminos and Geiss) that are not derived equivalent and gives a partial positive answer to a conjecture due to Bobiński and Malicki on gentle 2-cycle algebras.     

JORDAN–KRONECKER INVARIANTS OF FINITE-DIMENSIONAL LIE ALGEBRAS

For any finite-dimensional Lie algebra we introduce the notion of Jordan–Kronecker invariants, study their properties, and discuss examples. These invariants naturally appear in the framework of the bi-Hamiltonian approach to integrable systems on Lie algebras and are closely related to Mischenko–Fomenko’s argument shift method. We also state a generalised argument shift conjecture and prove it for many series of Lie algebras.     

On the Melvin–Morton–Rozansky conjecture

We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander–Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the “colored” Jones polynomial). We first reduce the problem to the level of weight systems using a general principle, which may be of some independent interest, and which sometimes allows to deduce equality of Vassiliev invariants from the equality of their weight systems. We then prove the conjecture combinatorially on the level of weight systems. Finally, we prove a generalization of the Melvin–Morton–Rozansky (MMR) conjecture to knot invariants coming from arbitrary semi-simple Lie algebras. As side benefits we discuss a relation between the Conway polynomial and immanants and a curious formula for the weight system of the colored Jones polynomial. Oblatum 28-VII-1994 & 5-XI-1995 & 22-XI-1995     

Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture

Let A and B be two finite dimensional algebras over an algebraically closed field, related to each other by a stable equivalence of Morita type. We prove that A and B have the same number of isomorphism classes of simple modules if and only if their 0-degree Hochschild Homology groups HH ₀(A) and HH ₀(B) have the same dimension. The first of these two equivalent conditions is claimed by the Auslander-Reiten conjecture. For symmetric algebras we will show that the Auslander-Reiten conjecture is equivalent to other dimension equalities, involving the centers and the projective centers of A and B. This motivates our detailed study of the projective center, which now appears to contain the main obstruction to proving the Auslander-Reiten conjecture for symmetric algebras. As a by-product, we get several new invariants of stable equivalences of Morita type.     

Application of classical invariant theory to biholomorphic classification of plane curve singularities,and associated binary forms

We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a method for extracting a complete set of invariants of homogeneous plane curve singularities from their moduli algebras.     

A Hilbert-Nagata theorem in noncommutative invariant theory

Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relatively free algebra is finitely generated for any group action from the class of Nagata. In particular, in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in Lie sense. As a consequence we extend the Hilbert-Nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is Lie nilpotent. We also prove that the Hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non-matrix polynomial identity is rational, if the action satisfies the condition of Nagata.

     

Special subvarieties of EPW sextics

We study the geometry of EPW sextics in order to produce special subvarieties. In particular we exhibit a (singular) Enriques surface and we compute its class in the Chow ring of the sextic. In order to do this, we produce an explicit degeneration of double EPW sextics to a Hilbert scheme of two points on a quartic surface, both in the smooth and in the singular case (keeping the singularities). The fixed locus of the covering involution on the double EPW sextic degenerates to the surface of bitangents to the quartic, which can be shown to be birational to an Enriques surface, provided the quartic acquires enough nodes. This construction is used in Ferretti (Algebra Number Theory, 2009a) as a starting point to prove a conjecture of Beauville and Voisin on the Chow ring of irreducible symplectic varieties, in the particular case of a very general double EPW sextic.     

Vinberg’s θ-groups in positive characteristic and Kostant–Weierstrass slices

We generalize the basic results of Vinberg’s θ-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard) Weyl group. We deduce that the ring of invariants associated to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded Lie algebra (in zero or odd positive characteristic), confirming a conjecture of Popov in this case.     

Enumeration of algebras close to absolutely free algebras and binary trees

We study three classes of algebras: absolutely free algebras, free commutative non-associative, and free anti-commutative non-associative algebras. We study asymptotics of the growth for free algebras of these classes and for their subvarieties as well. Mainly, we study finitely generated algebras, also the codimension growth for varieties in theses classes is studied. For these purposes we use ordinary generating functions as well as exponential generating functions. The following subvarieties are studied in these classes: solvable, completely solvable, right-nilpotent, and completely right-nilpotent subvarieties. The obtained results are equivalent to an enumeration of binary labeled and unlabeled rooted trees that do not contain some forbidden subtrees. We enumerate these trees using generating functions. For solvable and right-nilpotent algebras the generating functions are algebraic. For completely solvable and completely right-nilpotent algebras the respective functions are rational. It is known that these three varieties of algebras satisfy Schreier's property, i.e., subalgebras of free algebras are free. For free groups, there is Schreier's formula for the rank of a subgroup of a free group. We find analogues of this formula for these varieties. They are written in terms of series. As an application, we study invariants of finite groups acting on absolutely free algebras.     

Wheels, wheeling, and the Kontsevich integral of the Unknot

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of uni-trivalent diagrams. The two formulas use the related notions of “Wheels” and “Wheeing”. We prove these formulas ‘on the level of Lie algebras’ using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras. In a brief epilogue we report on recent proofs of our full conjectures, by Kontsevich [Ko2] and by DBN, DPT, and T. Q. T. Le, [BLT]. This paper is available electronically     

Algebraic isomorphisms of limit algebras

We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider the consequences of this result. In particular, we give partial solutions to a conjecture and an open problem by Power. As a further consequence, we describe epimorphisms between various classes of limit algebras.

     

The completeness of the isomorphism relation for countable Boolean algebras

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen's classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF -algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.

     

On the finiteness of differential invariants

In this paper we show how Weil's theory of near points yields a new light on the classical approaches to the study of the differential invariants of a sheaf of tangent vector fields. We give conditions for the existence of invariant derivations for a sheaf of tangent vector fields, which allows to apply Lie's algorithm to obtain new differential invariants as quotients of Jacobian determinants of known ones. We give sufficient conditions for the asymptotic stability of the symbol of a sheaf of tangent vector fields and prove our main result, a finiteness theorem for the differential invariants of a sheaf of Lie algebras which simplifies and improves on the treatment given in J. Differential Geom. 10 (1975) 249-416.     

Universal central extension and the second invariant of homology of crossed modules in lie algebras

In this paper we show that the kernel of the universal central extension of a crossed module in Lie algebras is the second invariant of this crossed module. As a consequence of this result we obtain a recognition criterion for universal central extensions and a vanishing situation of two invariants associated to a crossed module in Lie algebras.     

Proof of a conjecture by Đoković on the Poincaré series of the invariants of a binary form

?okovi? (2006)  [3] gave an algorithm for the computation of the Poincaré series of the algebra of invariants of a binary form, where the correctness proof for the algorithm depended on an unproven conjecture. Here we prove this conjecture.     

Finitistic dimension and Igusa–Todorov algebras

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radical cube zero, monomial algebras and left serial algebras, etc. It is an open question whether all artin algebras are Igusa–Todorov. We provide some methods to construct many new classes of (2-)Igusa–Todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds. In particular, we show that the class of 2-Igusa–Todorov algebras is closed under taking endomorphism algebras of projective modules. Hence, if all quasi-hereditary algebras are 2-Igusa–Todorov, then all artin algebras are 2-Igusa–Todorov by [V. Dlab, C.M. Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1) (1989) 1–5] and have finite finitistic dimension.     

Combinatorial derived invariants for gentle algebras

We define derived equivalent invariants for gentle algebras, constructed in an easy combinatorial way from the quiver with relations defining these algebras. Our invariants consist of pairs of natural numbers and contain important information about the algebra and the structure of the stable Auslander-Reiten quiver of its repetitive algebra. As a by-product we obtain that the number of arrows of the quiver of a gentle algebra is invariant under derived equivalence. Finally, our invariants separate the derived equivalence classes of gentle algebras with at most one cycle.     

On the coinvariants of modular representations of cyclic groups of prime order

We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family of representations, we are able to describe the coinvariants despite the fact that an explicit generating set for the invariants is not known. In all cases our results confirm the conjecture of Harm Derksen and Gregor Kemper on degree bounds for generators of the Hilbert ideal. As an incidental result, we identify the coefficients of the monomials appearing in the orbit product of a terminal variable for the three-dimensional indecomposable representation.