Some extremal properties of the variety of Leibniz algebras left nilpotent of class at most three

It is proved, for the case in which the ground field is of characteristic zero, that the variety of Leibniz algebras left nilpotent of class at most three is a variety of almost exponential growth with almost polynomial growth of the colength and has almost finite multiplicities.     

Almost nilpotent varieties with non-integer exponents do exist

The existence of an almost nilpotent variety of linear algebras with noninteger exponent is proved. Examples of almost nilpotent varieties with integer exponents were only known so far.     

Generating functions for ternary algebras and ternary trees

The main object of study are ternary algebras, i.e., algebras with a trilinear operation. In this class we study finitely generated algebras and their growth, as well as the growth of codimensions of absolutely free algebras and some other varieties. For these purposes we use ordinary generating functions and exponential generating functions (the complexity functions). In the classes of absolutely free, free symmetric, free antisymmetric, and some other algebras we study left nilpotent and completely left nilpotent algebras and varieties. The obtained results are equivalent to the enumeration of ternary trees which contain no forbidden subtrees of a special kind. As the main result, we prove that the complexity functions of the varieties of completely left nilpotent and left nilpotent ternary algebras are algebraic.     

Infinite periodic words and almost nilpotent varieties

An almost nilpotent variety of linear growth is constructed in the paper for any infinite periodic word in an alphabet of two letters. A discrete series of different almost nilpotent varieties is also constructed. Only a few almost nilpotent varieties were studied previously and their existence was proved often under some additional assumptions. The existence of almost nilpotent varieties of arbitrary integer exponential growth with a fractional exponent is proved as well as the existence of a continual family of almost nilpotent varieties with not more than quadratic growth.     

The algebraic and geometric classification of nilpotent terminal algebras

We give algebraic and geometric classifications of 4-dimensional complex nilpotent terminal algebras. Specifically, we find that, up to isomorphism, there are 41 one-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, 18 two-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, 2 three-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, complemented by 21 additional isomorphism classes (see Theorem 13). The corresponding geometric variety has dimension 17 and decomposes into 3 irreducible components determined by the Zariski closures of a one-parameter family of algebras, a two-parameter family of algebras and a three-parameter family of algebras (see Theorem 15). In particular, there are no rigid 4-dimensional complex nilpotent terminal algebras.     

DEGENERATIONS OF 7-DIMENSIONAL NILPOTENT LIE ALGEBRAS

ABSTRACT

We study the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra laws. We classify all degenerations of (almost all) five-step and six-step nilpotent seven-dimensional complex Lie algebras. One of the main tools is the use of trivial and adjoint cohomology of these algebras. In addition, we give some new results on the varieties of complex Lie algebra laws in low dimension.     

Codimension Growth of Strong Lie Nilpotent Associative Algebras

We study Lie nilpotent varieties of associative algebras. We explicitly compute the codimension growth for the variety of strong Lie nilpotent associative algebras. The codimension growth is polynomial and found in terms of Stirling numbers of the first kind. To achieve the result we take the free Lie algebra of countable rank L(X), consider its filtration by the lower central series and shift it. Next we apply generating functions of special type to the induced filtration of the universal enveloping algebra U(L(X)) = A(X).     

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.     

The Classification of Non-Characteristically Nilpotent Filiform Leibniz Algebras

In this paper we investigate the derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension is decomposed into three non-intersected families. We found sufficient conditions under which filiform Leibniz algebras of the first family are characteristically nilpotent. Moreover, for the first family we classify non-characteristically nilpotent algebras by means of Catalan numbers. In addition, for the rest two families of filiform Leibniz algebras we describe non-characteristically nilpotent algebras, i.e., those filiform Leibniz algebras which lie in the complementary set to those characteristically nilpotent.     

Necessary and sufficient conditions of polynomial growth of varieties of Leibniz-Poisson algebras

Leibniz-Poisson algebras are generalizations of Poisson algebras. We give equivalent conditions of polynomial growth of a variety of Leibniz-Poisson algebras over a field of characteristic zero. We find all varieties of Leibniz-Poisson algebras with almost polynomial growth belonging to a certain class of varieties.     

Minimal varieties of algebras of exponential growth

The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of given exponent and of finite basic rank. As a consequence, we describe the corresponding T-ideals of the free algebra and we compute the asymptotics of the related codimension sequences, verifying in this setting some known conjectures. We also show that the number of these minimal varieties is finite for any given exponent. We finally point out some relations between the exponent of a variety and the Gelfand-Kirillov dimension of the corresponding relatively free algebras of finite rank.     

Correlation of Poisson algebras and Lie algebras in the language of identities

In the paper, the varieties of Poisson algebras whose ideals of identities contain the identity {x, y}· {z, t} = 0 are studied, and the correlation of these varieties with varieties of Lie algebras is investigated. A variety of Poisson algebras of almost exponential growth is presented. An example of a variety of Poisson algebras with fractional exponent is also given.     

Equivalent conditions of polynomial growth of a variety of Poisson algebras

Equivalent conditions of the polynomial codimension growth of a variety of Poisson algebras over a field of characteristic zero are presented and it is shown that there are only two varieties of Poisson algebras with almost polynomial growth.     

Singular Poisson-Kähler geometry of Scorza varieties and their secant varieties

Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kähler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.     

Locally finite Lie algebras¶with unitary highest weight representations

In this paper we essentially classify all locally finite Lie algebras with an involution and a compatible root decomposition which permit a faithful unitary highest weight representation. It turns out that these Lie algebras have many interesting relations to geometric structures such as infinite-dimensional bounded symmetric domains and coadjoint orbits of Banach–Lie groups which are strong K?hler manifolds. In the present paper we concentrate on the algebraic structure of these Lie algebras, such as the Levi decomposition, the structure of the almost reductive and locally nilpotent part, and the structure of the representation of the almost reductive algebra on the locally nilpotent ideal. Received: 2 August 2000 / Revised version: 10 January 2001     

Lie algebras with almost dimensionally nilpotent inner derivations

The structure of the Lie algebras with almost dimensionally nilpotent inner derivations is studied. It is proved that, if the base field is of characteristic 0, then, when d>6 is odd, there exist just two d-dimensional Lie algebras; when d>6 is even, there exists just one d-dimensional Lie algebra such that these Lie algebras are nonsolvable and have some almost dimensionally nilpotent inner derivations.     

Jordan recoverability of some subcategories of modules over gentle algebras

Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowroński in the 1980s. Modules over such an algebra can be described by string and band combinatorics in the associated gentle quiver from the work of M.C.R. Butler and C.M. Ringel. Any module can be naturally associated to a quiver representation. A nilpotent endomorphism of a quiver representation induces linear transformations over vector spaces at each vertex. Generically among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. We focus on subcategories additively generated by all the indecomposable representations of a gentle quiver, including a fixed vertex in their support. We show a characterization of the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.     

On Ricci negative solvmanifolds and their nilradicals

In the homogeneous case, the only curvature behavior which is still far from being understood is Ricci negative. In this paper, we study which nilpotent Lie algebras admit a Ricci negative solvable extension. Different unexpected behaviors were found. On the other hand, given a nilpotent Lie algebra, we consider the space of all the derivations such that the corresponding solvable extension has a metric with negative Ricci curvature. Using the nice convexity properties of the moment map for the variety of nilpotent Lie algebras, we obtain a useful characterization of such derivations and some applications.     

Radical cube zero selfinjective algebras of finite complexity

One of our main results is a classification of all the possible quivers of selfinjective radical cube zero finite-dimensional algebras over an algebraically closed field having finite complexity. In the paper (Erdmann and Solberg, 2011) [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite-dimensional algebra to have such a theory of support varieties implies that the algebra has finite complexity. Hence this paper is a partial extension of [5].     

Exponents of varieties of lie algebras with a nilpotent commutator subalgebra

We prove that the growth exponent of any variety of Lie algebras with a nilpotent commutator subalgebra is integral.