Lie algebras and algebras of associative type

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕ _αεG A _α graded by some group G, over a field of characteristic zero, has a nonzero component A ₁ (where 1 stands for the identity element of G), and A ₁ is a semisimple associative algebra. Let B = ⊕ _αεG B _α be a finite-dimensional semisimple Lie algebra over a prime field F _p , and let B be graded by a commutative group G. If B = F _p ? _? A _L , where A _L is the commutator algebra of a ?-algebra A = ⊕ _αεG A _α ; if ? ? _? A is an algebra of associative type, then the 1-component of the algebra K ? _? B, where K stands for the algebraic closure of the field F _p , is the sum of some algebras of the form gl(n _i ,K).     

Lie gradings on associative algebras

In this paper we apply the method of functional identities to the study of group gradings by an abelian group G on simple Lie algebras, under very mild restrictions on the grading group or the base field of coefficients.     

On generalized Lie derivations of Lie ideals of prime algebras

Let A be a prime algebra of characteristic not 2 with extended centroid C, let R be a noncentral Lie ideal of A and let B be the subalgebra of A generated by R. If f,d:R→A are linear maps satisfying that

     

On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern     

Lie ideals of graded associative algebras

We study the Lie structure of graded associative algebras. Essentially, we analyze the relation between Lie and associative graded ideals, and between Lie and associative graded derivations. Gathering together results on both directions, we compute maximal graded algebras of quotients of graded Lie algebras that arise from associative algebras. We also show that the Lie algebra Der _gr (A) of graded derivations of a graded semiprime associative algebra is strongly non-degenerate (modulo a certain ideal containing the center of Der _gr (A)).     

On the duality of generalized Lie and Hopf algebras

We show how, under certain conditions, an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of Hopf algebras. This leads us to an abstract version of Michaelis' theorem, stating that given a Hopf algebra H  , there is a natural isomorphism of Lie algebras Q(H)^?≅P(H^°)$Q{(H)}^{?}\cong P(H^{°})$, where Q(H)^?$Q{(H)}^{?}$ is the dual Lie algebra of the Lie coalgebra of indecomposables of H  , and P(H^°)$P(H^{°})$ is the Lie algebra of primitive elements of the Sweedler dual of H. We apply our theory to Turaev's Hopf group-(co)algebras.     

Invariants of Lie superalgebras acting on associative algebras

LetR be a semiprime algebra over a fieldK acted on by a finite-dimensional Lie superalgebraL. The purpose of this paper is to prove a series of going-up results showing how the structure of the subalgebra of invariantsR ^Lis related to that ofR. Combining several of our main results we have: Theorem: Let R be a semiprime K-algebra acted on by a finite-dimensional nilpotent Lie superalgebra L such that if characteristic K=p then L is restricted and if characteristic, K=0 then L acts on R as algebraic derivations and algebraic superderivations.

1. If R^L is right Noetherian, then R is a Noetherian right R^L-module. In particular, R is right Noetherian and is a finitely generated right R^L-module.
2. If R^L is right Artinian, then R is an Artinian right R^L-module. In particular, R is right Artinian and is a finitely generated right R^L-module.
3. If R^L is finite-dimensional over K then R is also finite-dimensional over K.
4. If R^L has finite Goldie dimension as a right R^L-module, then R has finite Goldie dimension as a right R-module.
5. If R^L has Krull dimension α as a right R^L-module, then R has Krull dimension α as a right R^L-module. Thus R has Krull dimension at most α as a right R-module.
6. If R is prime and R^L is central, then R satisfies a polynomial identity.
7. If L is a Lie algebra and R^L is central, then R satisfies a polynomial identity.

We also provide counterexamples to many questions which arise in view of the results in this paper.     

Serre Lie algebras of generalized Jacobians

In this work we construct an example of a generalized Jacobian of an elliptic curve defined over a field of algebraic numbers k such that the Serre Lie algebra p-adic representation of the Galois group of the algebraic closure of the field k in its Tate module is irreducible.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 571–576, April, 1976.In conclusion the authors thank O. N. Vvedenskii for guidance in this work.     

Lie,Jordan and proper codimensions of associative algebras

We study the exponential rate of growth of the sequence of proper, Lie and Jordan codimensions of an associative algebra. We show that for any finite dimensional associative algebra, the exponential rates of growth can be explicitly computed and are strictly related to the PI-exponent of the algebra. The first author was partially supported by MIUR of Italy. The second author was partially supported by RFBR grant No 06-01-00485 and SSC-5666.2006.1     

Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras

This paper studies the concepts of a totally compatible dialgebra and a totally compatible Lie dialgebra,defined to be a vector space with two binary operations that satisfy individual and mixed associativity conditions and Lie algebra conditions respectively.We show that totally compatible dialgebras are closely related to bimodule algebras and semi-homomorphisms.More significantly,Rota-Baxter operators on totally compatible dialgebras provide a uniform framework to generalize known results that Rota-Baxter related operators give tridendriform algebras.Free totally compatible dialgebras are constructed.We also show that a Rota-Baxter operator on a totally compatible Lie dialgebra gives rise to a PostLie algebra,generalizing the fact that a Rota-Baxter operator on a Lie algebra gives rise to a PostLie algebra.