On Some Ideals in the Free Nonassociative Algebra

The free Lie, right, and left Leibniz algebras are obtained as quotients of the free nonassociative algebra by suitable ideals. In this article, we prove some remarkable properties of these ideals.     

Modules Whose Endomorphism Rings are Division Rings

The well-known Schur's Lemma states that the endomorphism ring of a simple module is a division ring. But the converse is not true in general. In this paper we study modules whose endomorphism rings are division rings. We first reduce our consideration to the case of faithful modules with this property. Using the existence of such modules, we obtain results on a new notion which generalizes that of primitive rings. When R is a full or triangular matrix ring over a commutative ring, a structure theorem is proved for an R-module M such that End _R (M) is a division ring. A number of examples are given to illustrate our results and to motivate further study on this topic.     

Classification of Some Graded not Necessarily Associative Division Algebras I

We study not necessarily associative (NNA) division algebras over the reals. We classify in this paper series those that admit a grading over a finite group G, and have a basis {v _g |g ∈ G} as a real vector space, and the product of these basis elements respects the grading and includes a scalar structure constant with values only in {1, ? 1}. We classify here those graded by an abelian group G of order |G| ≤8 with G non–isomorphic to ?/8?. We will find the complex, quaternion, and octonion algebras, but also a remarkable set of novel non–associative division algebras.     

Embedding Ordered Valued Domains into Division Rings

We study some classes of ordered domains that are embeddable in division rings. We prove the ordered version of the Cohn–Lichtman embedding theorem for valued domains. A question of Glass is answered in the negative. Furthermore, we prove that universal enveloping algebras of Lie algebras over formally real fields can be embedded into ordered division rings. The author acknowledges the financial support from the state budget by the Slovenian Research Agency (project no. Z1-9570-0101-06).     

Dimension Formulas for the Free Nonassociative Algebra

The free nonassociative algebra has two subspaces which are closed under both the commutator and the associator: the Akivis elements and the primitive elements. Every Akivis element is primitive, but there are primitive elements which are not Akivis. Using a theorem of Shestakov, we give a recursive formula for the dimension of the Akivis elements. Using a theorem of Shestakov and Umirbaev, we prove a closed formula for the dimension of the primitive elements. These results generalize the Witt dimension formula for the Lie elements in the free associative algebra.     

Free orbit dimension of finite von Neumann algebras

We introduce a new free entropy invariant, which yields improvements of most of the applications of free entropy to finite von Neumann algebras, including those with Cartan subalgebras, simple masas, property T, property Γ, nonprime factors, and thin factors.     

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M _n ) of a free metabelian Lie algebra M _n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M ₄) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M ₃) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.     

Free Ornstein-Uhlenbeck processes

Free Ornstein-Uhlenbeck processes are studied in finite von Neumann algebras. It is shown that a free self-decomposable probability measure on R can be realized as the distribution of a stationary free Ornstein-Uhlenbeck process driven by a free Levy process. A characterization of a probability measure on R to be the stationary distribution of a periodic free Ornstein-Uhlenbeck process driven by a free Levy process is given in terms of the Levy measure of the measure. Finally, the notion of a free fractional Brownian motion is introduced. It is proved that the free stochastic differential equation driven by a fractional free Brownian motion has a unique solution. We call the solution a fractional free Ornstein-Uhlenbeck process.     

Embedding Problems of Division Algebras

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann, and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.     

Metrically Universal Generic Structures in Free Amalgamation Classes

We prove that each ∀₁ free amalgamation class K over a finite relational language L admits a countable generic structure M isometrically embedding all countable structuresin K relative to a fixed metric. We expand L by infinitely many binary predicates expressingdistance, and prove that the resulting expansion of K has a model companion axiomatizedby the first‐order theory of M. The model companion is non‐finitely axiomatizable, evenover a strong form of the axiom scheme of infinity.     

A Note on Free Groups in the Ring of Fractions of Skew Polynomial Rings

Let L be a function field over the rationals and let D denote the skew field of fractions of L[t;σ], the skew polynomial ring in t, over L, with automorphism σ. We prove that the multiplicative group D ^× of D contains a free noncyclic subgroup.     

The Structures of the Cohomology Rings of Wreathed 2-Groups

Let S be a wreathed 2-group. In this article we construct a free ?S-resolution of ? and describe the ring structure of the cohomology of wreathed 2-group S.     

Free cumulants,Schröder trees,and operads

The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Faà di Bruno algebra, and then to the group of a free operad over Schröder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.     

A Problem of B. Plotkin for S-Acts: Automorphisms of Categories of Free S-Acts

In this article, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings. We use this result to compute the Gorenstein global dimension of some particular cases of trivial extensions of rings and of group rings.     

Notes on Free Monadic Boolean Algebras

If F B(2 ⁿ – 1) denotes the Boolean algebra with 2 ⁿ – 1 free generators and P(2 ⁿ ) is the Cartesian product of 2 ⁿ Boolean algebras all equal to F B(2 ⁿ – 1), we define on P(2 ⁿ ) an existential quantifier by means of a relatively complete Boolean subalgebra of P(2 ⁿ ) and we prove that (P(2ⁿ),) is the monadic Boolean algebra with n free generators. Every element of P(2 ⁿ ) is a 2 ⁿ -tuple whose coordinates are in F B(2 ⁿ – 1); in particular, so are the n generators of P(2 ⁿ ). We indicate in this work the coordinates of the n generators of P(2 ⁿ ).     

On Ordered Division Rings

Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel's axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under x xa ² for nonzero a, instead of requiring that positive elements have a positive product. In this work, this type of ordering is studied in the case of a division ring. It is shown that it actually behaves the same as in the commutative case. Further, it is shown that the bounded subring associated with that ordering is a valuation ring which is preserved under conjugation, so one can associate a natural valuation to a semiordering.