Finite-dimensional Jordan algebras admitting the structure of a Jordan bialgebra

We define the concepts of a triangular and a quasitriangular Jordan bialgebras. It is proved that a finite-dimensional Jordan algebra J over an algebraically closed field Φ admits the structure of a quasitriangular Jordan bialgebra with nonzero comultiplication, provided that J is not a direct sum of fields, algebras H(Φ₂) and H(Φ₃), null extensions of Φ, and of algebras with zero multiplication. Supported by RFFR grant No. 98-01-01142. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 40–67, January–February, 1999.     

The Kantor-Koecher-Tits construction for Jordan coalgebras

The relationship between Jordan and Lie coalgebras is established. We prove that from any Jordan coalgebra 〈L(A), Δ〉, it is possible to construct a Lie coalgebra 〈L(A), Δ_L〉. Moreover, any dual algebra of the coalgebra 〈L(A), Δ_L〉 corresponds to a Lie algebra that can be determined from the dual algebra for (A, Δ), following the Kantor-Koecher-Tits process. The structure of subcoalgebras and coideals of the coalgebra 〈L(A), Δ_L〉 is characterized. Supported by ISF grant No. RB 6000. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 173–189, March–April, 1996.     

Bidendriform bialgebras, trees, and free quasi-symmetric functions

We introduce bidendriform bialgebras, which are bialgebras such that both product and coproduct can be split into two parts satisfying good compatibilities. For example, the Malvenuto-Reutenauer Hopf algebra and the non-commutative Connes-Kreimer Hopf algebras of planar decorated rooted trees are bidendriform bialgebras. We prove that all connected bidendriform bialgebras are generated by their primitive elements as a dendriform algebra (bidendriform Milnor-Moore theorem) and then is isomorphic to a Connes-Kreimer Hopf algebra. As a corollary, the Hopf algebra of Malvenuto-Reutenauer is isomorphic to the Connes-Kreimer Hopf algebra of planar rooted trees decorated by a certain set. We deduce that the Lie algebra of its primitive elements is free in characteristic zero (G. Duchamp, F. Hivert and J.-Y. Thibon conjecture).     

Lie bialgebras arising from alternative and Jordan bialgebras

We consider connection between simple alternative D-bialgebras and Lie bialgebras. We prove that each finite-dimensional alternative noncommutative algebra over an algebraically closed field may be equipped with the nontrivial structure of an alternative D-bialgebra.     

The homotopy theory of bialgebras over pairs of operads

We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in two steps. In the first step, we equip coalgebras over an operad with a cofibrantly generated model category structure. In the second step we use the adjunction between bialgebras and coalgebras via the free algebra functor. This result allows us to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras in chain complexes.     

Lie bialgebras of generalized Witt type

In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W(?)W) is trivial.     

Witt型与Virasoro型Lie双代数的对偶

本文给出单边的Witt 型、Witt 型和Virasoro 型Lie 双代数的对偶Lie 双代数结构. 由此, 本文得到一系列无限维Lie 代数.     

Dual Coalgebras of Jordan Bialgebras and Superalgebras

W. Michaelis showed for Lie bialgebras that the dual coalgebra of a Lie algebra is a Lie bialgebra. In the present article we study an analogous question in the case of Jordan bialgebras. We prove that a simple infinite-dimensional Jordan superalgebra of vector type possesses a nonzero dual coalgebra. Thereby, we demonstrate that the hypothesis formulated by W. Michaelis for Lie coalgebras fails in the case of Jordan supercoalgebras.     

Types of action of Lie ε-algebras

We introduce the concept of a type of action. Various types of group actions and actions of Lie ε-algebras are examined. The main result is the classification of types of action F of Lie ε-algebras with the property that for all Lie ε-algebras L, F(L), as an algebra, is the bozonization of L. Supported by RFFR grant No. 93-01-16171 and by ISF grants RPS000 and RPS300. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 468–475, July–August, 1996.     

Novikov-Poisson algebras and associative commutative derivation algebras

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras. Supported by RFBR (grant No. 05-01-00230), by SB RAS (Integration project No. 1.9), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 186–202, March–April, 2008.     

Compatibility of quantization functors of Lie bialgebras with duality and doubling operations

We study the behavior of the Etingof–Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are “compatible with duality”, i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof–Kazhdan quantizations of some affine Lie superalgebras coincide with their Drinfeld–Jimbo-type quantizations. To the memory of Paulette Libermann (1919–2007)     

Quantization of Lie bialgebras, II

This paper is a continuation of [EK]. We show that the quantization procedure of [EK] is given by universal acyclic formulas and defines a functor from the category of Lie bialgebras to the category of quantized universal enveloping algebras. We also show that this functor defines an equivalence between the category of Lie bialgebras over k [[h]] and the category of quantized universal enveloping (QUE) algebras.     

Lie rings admitting an automorphism of order 4 with few fixed points

Lie rings that admit an automorphism of order 4 with few fixed points are considered. For a Lie ring (algebra) L admitting an automorphism of order 4 with a finite number m of fixed points (with a finite-dimensional subalgebra of fized points of dimension m), it is proved that the subring 4L (algebra L) contains an ideal M with a subring of m-bounded index in the additive group of M (a subalgebra of m-bounded codimension), which is nilpotent of class bounded by some constant. It is also shown that, under the same premise, the factor-ring 4L/M (factor-algebra L/M) contains a subring of m-bounded index in the additive group of 4L/M (a subalgebra of m-bounded codimension), which is nilpotent of class ≤2. Moreover, L has a subring of m-bounded index in the additive group of L (a subalgebra of m-bounded codimension), which is soluble of derived length bounded by a constant. Supported by RFFR grant No. 94-01-00048 and by ISF grant NQ7000. Translated fromAlgebra i Logika, Vol. 35, No. 1, pp. 41–78, January–February, 1996.     

Purely Hom-Lie bialgebras

In this paper, we first show that there is a Hom-Lie algebra structure on the set of(σ, σ)-derivations of an associative algebra. Then we construct the dual representation of a representation of a Hom-Lie algebra.We introduce the notions of a Manin triple for Hom-Lie algebras and a purely Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a one-to-one correspondence between Manin triples for Hom-Lie algebras and purely Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and construct solutions of the classical Hom-Yang-Baxter equations in some special Hom-Lie algebras using Hom-O-operators.     

Coassociative magmatic bialgebras and the Fine numbers

We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n−2 operations of arity n. The dimension of the space of all the n-ary operations of this primitive operad turns out to be the Fine number F _n−1. In short, the triple of operads (As, Mag, MagFine) is good. The third author work is partially supported by FONDECYT Project 1060224     

Noncommutative invariants of bialgebras

Suppose that H is a bialgebra over a field C and R = CV is the tensor algebra of the C-space V endowed with the structure of an H-module algebra, so that V is a submodule of the H-module R, R^H is the algebra of H-invariants, and W, the support of the algebra R^H, is the smallest subspace of the C-space V such that . The main result of the paper is the theorem stating that if the algebra of H-invariants R^H is finitely generated, then the support of R^H is a finite-dimensional submodule of the H-module V, whose elements are H-semi-invariants of the same weight.Translated fromAlgebra i Logika, Vol. 33, No. 6, pp. 654–680, November–December, 1994.Supported by the Soros Foundation (grant RPS000) and the Russian Foundation for Fundamental Research (grant No. 93-01-16171).     

Derivation of prime rings of positive characteristic

Let L be a finite-dimensional differential Lie algebra acting on a prime ring R and let the inner part {ie49-1} of L be quasi-Frobenius. Then a constant ring R^L is prime iff {ie49-2} is a differentially simple ring. A ring of constants is semiprime iff {ie49-3} is a direct sum of differentially simple rings, and the prime dimension of a constant ring is equal to the number of differentially simple summands {ie49-4}. The Galois closure of L is obtained from L by adding all the inner derivations of a symmetric Martindale quotient ring which agree with elements from {ie49-5}. Supported by RFFR grant No. 93-01-16171 and by ISF grant RPS000-RPS300. Translated fromAlgebra i Logika, Vol. 35, No. 1, pp. 88–104, January–February, 1996.     

δ-Derivations of simple finite-dimensional Jordan superalgebras

We describe non-trivial δ-derivations of semisimple finite-dimensional Jordan algebras over an algebraically closed field of characteristic not 2, and of simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic 0. For these classes of algebras and superalgebras, non-zero δ-derivations are shown to be missing for δ ≠ 0, 1/2, 1, and we give a complete account of 1/2-derivations. Supported by RFBR grant No. 05-01-00230 and by RF Ministry of Education and Science grant No. 11617. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 585–605, September–October, 2007.     

The Chevalley and Costant theorems for Mal’tsev algebras

Centers of universal envelopes for Mal’tsev algebras are explored. It is proved that the center of the universal envelope for a finite-dimensional semisimple Mal’tsev algebra over a field of characteristic 0 is a ring of polynomials in a finite number of variables equal to the dimension of its Cartan subalgebra, and that universal enveloping algebra is a free module over its center. Centers of universal enveloping algebras are computed for some Mal’tsev algebras of small dimensions. Supported by FAPESP grant No. 04/08537-4 and by SO RAN grant No. 1.9. Supported by FAPESP grant Nos. 05/60142-7, 05/60337-2 and by CNPq grant No. 304991/2006-6. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 560–584, September–October, 2007.