On structure of cluster algebras of geometric type Ⅰ:In view of sub-seeds and seed homomorphisms

Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point of gluing of seeds. As an application, for(rooted) cluster algebras, we completely classify rooted cluster subalgebras and characterize rooted cluster quotient algebras in detail. Also,we build the relationship between the categorification of a rooted cluster algebra and that of its rooted cluster subalgebras. Note that cluster algebras of geometric type studied here are of the sign-skew-symmetric case.     

Cluster tilting for tilted algebras

We build a connection between iterated tilted algebras with trivial cluster tilting subcategories and tilted algebras of finite type.Moreover,all tilted algebras with cluster tilting subcategories are determined in terms of quivers.As a result,we draw the quivers of Auslander’s 1-Gorenstein algebras with global dimension 2 admitting trivial cluster tilting subcategories,which implies that such algebras are of finite type but not necessarily Nakayama.     

On Limits of Mishchenko--Fomenko Subalgebras in Poisson Algebras of Semisimple Lie Algebras

In this paper, the commutative (with respect to the Poisson bracket) subalgebras in the Poisson algebras of the semisimple Lie algebras are considered on condition that these subalgebras are limits of Mishchenko--Fomenko subalgebras. We study the case of the degeneration within a fixed Cartan subalgebra. The structure of the limit subalgebras is described (i.e., it is proved that these subalgebras are free, and their generators are found). The classification of the limit subalgebras of the above type is also established.     

Quantized Weyl algebras at roots of unity

We classify the centers of the quantized Weyl algebras that are polynomial identity algebras and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are given: one based on Poisson geometry and deformation theory, and the other using techniques from quantum cluster algebras. Furthermore, we classify the PI quantized Weyl algebras that are free over their centers and prove that their discriminants are locally dominating and effective. This is applied to solve the automorphism and isomorphism problems for this family of algebras and their tensor products.     

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.     

Equivalences for blocks of the Weyl groups

We describe the source algebras of the blocks of the Weyl groups of type B and type D in terms of the source algebras of the blocks of the symmetric groups. As a consequence, we show that Puig's conjecture on the finiteness of the number of isomorphism classes of source algebras for blocks of finite groups with a fixed defect group holds for these classes of groups. We also show how certain isomorphisms between subalgebras of block algebras of the symmetric groups can be lifted to block algebras of the Weyl groups of type B.

     

Lattice definability of semisimple jordan algebras

In this paper we prove that two finite-dimensional linear Jordan algebras over an algebraically closed field with isothermic lattices of subalgebras must bi isothemic if one of them is semisimple non-isothermic to F. As a corollary of this fact, we prove that two unital Jordan algebras with isothermic lattices of subalgebras must have the same dimension when the ground field is algebraically closed of characteristic zero. Through this work we see similar results in more general fields for particular families of simple Jordan algebras.     

Partially-2-Homogeneous Monounary Algebras

This paper is a continuation of [5], where k-homogeneous and k-set-homogeneous algebras were defined. The definitions are analogous to those introduced by Fraïssé [2] and Droste, Giraudet, Macpherson, Sauer [1] for relational structures. In [5] we found all 2-homogeneous and all 2-set-homogeneous monounary algebras when the homogenity is considered with respect to subalgebras, to connected subalgebras and with respect to connected partial subalgebras, respectively. The results of [3], where all homogeneous monounary algebras were characterized, were applied in [4] for 1-homogeneity.The aim of the present paper is to describe all monounary algebras which are 2-homogeneous and 2-set-homogeneous with respect to partial subalgebras, respectively; we will say that they are partially-2-homogeneous and partially-2-set-homogeneous.     

On abelian subalgebras and ideals of maximal dimension in supersolvable Lie algebras

In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not 2. Throughout the paper, we also give several examples to clarify some results.     

The Ideal Index of Subalgebras of a Finite Dimensional Lie Algebra

D. A. Towers introduced the notion of ideal index of a maximal subalgebra of a Lie algebra, and used it to analyze the influence of maximal subalgebras on the structure of a finite dimensional Lie algebras.

In this article, we generalize the ideal index from maximal subalgebras to all subalgebras, and obtain some new characterizations of solvable and supersolvable Lie algebras by the ideal indices of some certain subalgebras.     

On some constructions of algebraic objects

Mono-unary algebras may be used to construct homomorphisms, subalgebras, and direct products of algebras of an arbitrary type.     

Property ∏σ and tensor products of operator algebras

In this paper, we introduce Property ∏σ of operator algebras and prove that nest subalgebras and the finite-width CSL subalgebras of arbitrary von Neumann algebras have Property ∏σ.Finally, we show that the tensor product formula alg ML1-(×)algNL2 = algM-(×)N(L1 (×) L2) holds for any two finite-width CSLs L1 and L2 in arbitrary von Neumann algebras M and N, respectively.     

Positivity for cluster algebras from surfaces

We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.     

On generators and relations for certain involutory subalgebras of kac-moody lie algebras ∗

We find generators and relations for those subalgebras of Kac-Moody Lie algebras that are the fixed point algebras of certain involutions. Specifically the involution must involve the Cartan involution which interchanges the positive and negative generators. We go on to apply these results to the G.I.M. algebras, which were introduced as natural generalizations of Kac-Moody algebras by P. Slodowy. We show such algebras are isomorphic to subalgebras of Kac-Moody algebras. From this we are able to derive someinteresting interrelations between certain Kac-Moody algebras.     

Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid

We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving and order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the above map is bijective. In the special case of the Borel part of quantized enveloping algebras our result implies a conjecture of Kharchenko.     

Some properties of semifinite tracial subalgebras

We obtained some characterizations of Jensen type inequalities in tracial subalgebras and gave some characterizations of subdiagonal algebras of semifinite von Neumann algebras.     

Subalgebras of FA-presentable algebras

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First, an example is given to show that the class of finitely generated FA-presentable algebras is not closed under forming finitely generated subalgebras, even within the class of algebras with only unary operations. In contrast, a finitely generated subalgebra of an FA-presentable algebra with a single unary operation is itself FA-presentable. Furthermore, it is proven that the class of unary FA-presentable algebras is closed under forming finitely generated subalgebras and that the membership problem for such subalgebras is decidable.     

On classical artinian localizations

ABSTRACT

We present some results about Lie algebras, which can be written as the sum of two subalgebras in two cases: where both subalgebras are simple or both are nilpotent. In the first case we suggest new examples of simple Lie algebras admitting decomposition into the sum of simple subalgebras and give explicit realizations where the existence of such decompositions was established earlier. We single out cases where such decomposition is not possible. We also construct examples of solvable Lie algebras, which are the sums of two nilpotent subalgebras, and the derived length of the sum is greater than the sum of the nilpotent indexes of the summands.