Study of an identity

We solve the word problem of the identity x(yz) = (xy)(yz) by investigating a certain group describing the geometry of that identity. We also construct a concrete realization of the one-generated free algebra relative to the above identity. Received March 23, 2001; accepted in final form July 6, 2002.     

关于扭Hopf代数的一些注记

In this paper,we get some properties of the antipode of a twisted Hopf algebra.We proved that the graded global dimension of a twisted Hopf algebra coincides with the graded projective dimension of its trivial module k,which is also equal to the projective dimension of k.     

Gelfand–Kirillov Dimension for Automorphism Groups of Relatively Free Algebras

Using ideas of our recent work on automorphisms of residually nilpotent relatively free groups, we introduce a new growth function for subgroups of the automorphism groups of relatively free algebras Fⁿ(V) over a field of characteristic zero and the related notion of Gelfand-Kirillov dimension, and study their behavior. We prove that, under some natural restrictions, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is equal to the Gelfand-Kirillov dimension of the algebra Fⁿ(V). We show that, in some cases, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is smaller than the Gelfand-Kirillov dimension of the whole automorphism group, and calculate the Gelfand-Kirillov dimension of the automorphism group of Fⁿ(V) for some important varieties V.Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.2000 Mathematics Subject Classification: primary 16R10, 16P90; secondary 16W20, 17B01, 17B30, 17B40     

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.     

Down-Up Algebras From Trees

It is shown that the global dimension of any n-ary down-up algebra A _n  = A(n,α, β,γ) is less than or equal to n + 2, and when γ _i  = 0 for all i (A _n is graded by total degree in the generators), then the global dimension of A _n is n + 2. Furthermore, a sufficient condition for A _n to be prime is given; when γ _i  = 0 for all i this condition is also necessary. An example is given to show that the condition is not always necessary.     

Isomorphically rigid algebras

Various classes of non-associative algebras possessing the property of being rigid under abstract isomorphisms are studied. Supported by RFBR grant No. 06-01-00159a. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 483–502, July–August, 2007.     

Homological Dimension of Coalgebras and Crossed Coproducts

Let H be a Hopf k-algebra. We study the global homological dimension of the underlying coalgebra structure of H. We show that gl.dim(H) is equal to the injective dimension of the trivial right H-comodule k. We also prove that if D = C H is a crossed coproduct with invertible , then gl.dim(D) gl.dim(C) + gl.dim(H). Some applications of this result are obtained. Moreover, if C is a cocommutative coalgebra such that C ^* is noetherian, then the global dimension of the coalgebra C coincides with the global dimension of the algebra C ^*.     

On Lie p-algebras of cohomological dimension one

We prove that a Lie $p$-algebra of cohomological dimension one is one-dimensional, and discuss related questions.