Jordan (Super)Coalgebras and Lie (Super)Coalgebras

We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Koecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra (J,) with a dual algebra J ^*, there exists a Lie supercoalgebra (L ^c (J), _L ) whose dual algebra (L ^c (J))^* is the Lie KKT-superalgebra for the Jordan superalgebra J ^*. It is well known that some Jordan coalgebra J ⁰ can be constructed from an arbitrary Jordan algebra J. We find necessary and sufficient conditions for the coalgebra (L ^c (J ⁰),_L) to be isomorphic to the coalgebra (Loc(L _in (J)⁰), _L ⁰), where L _in (J) is the adjoint Lie KKT-algebra for the Jordan algebra J.     

T*-extension of a 3-Lie algebra

We introduce a new technique called T*-extension, of constructing a metric 3-Lie algebra out of an arbitrary 3-Lie algebra, and explore all possible metrics and corresponding signatures on this resulting metric 3-Lie algebra.     

Formality theorem for Lie bialgebras and quantization of twists and coboundary r-matrices

Let (g,δ?) be a Lie bialgebra. Let (U?(g),Δ?) a quantization of (g,δ?) through Etingof-Kazhdan functor. We prove the existence of a L_∞-morphism between the Lie algebra C(g)=Λ(g) and the tensor algebra (without unit) T₊U=T₊(U?(g)[−1]) with Lie algebra structure given by the Gerstenhaber bracket. When s is a twist for (g,δ), we deduce from the formality morphism the existence of a quantum twist F. When (g,δ,r) is a coboundary Lie bialgebra, we get the existence of a quantization R of r.     

Vertex coalgebras, comodules, cocommutativity and coassociativity

We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, D^∗, which hold on vertex coalgebras. The former two properties require grading. We then discuss comodule structure. We conclude by discussing instances where graded vertex coalgebras appear, particularly as related to Primc’s vertex Lie algebra and (universal) enveloping vertex algebras.     

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.     

Embedding of Jordan Copairs into Lie Coalgebras

Let (V, Δ) be a Jordan copair over a field Φ and let V? be its dual pair. Then there exists a Lie coalgebra (L ^c (V), Δ _L ) whose dual algebra (L ^c (V))? is the Kantor–Koecher–Tits construction for the pair V?. If Φ is a field of characteristic other than 2 or 3 then the Lie coalgebra (L ^c (J), Δ _L ) is locally finite-dimensional. As a corollary we derive that Jordan copairs over fields of characteristic other than 2 or 3 are locally finite-dimensional.     

The n-lie property of the Jacobian as a condition for complete integrability

We prove that an associative commutative algebra U with derivations D ₁, ..., D _n ε DerU is an n-Lie algebra with respect to the n-multiplication D ₁ ^ ? ^ D _n if the system {D ₁, ..., D _n } is in involution. In the case of pairwise commuting derivations this fact was established by V. T. Filippov. One more formulation of the Frobenius condition for complete integrability is obtained in terms of n-Lie multiplications. A differential system {D ₁, ..., D _n } of rank n on a manifold M ^m is in involution if and only if the space of smooth functions on M is an n-Lie algebra with respect to the Jacobian Det(D _i u _j ).     

On radicals of module coalgebras

We introduce the notion of an idempotent radical class of module coalgebras over a bialgebra B. We prove that if R is an idempotent radical class of B-module coalgebras, then every B-module coalgebra contains a unique maximal B-submodule coalgebra in R. Moreover, a B-module coalgebra C is a member of R if, and only if, DB is in R for every simple subcoalgebra D of C. The collection of B-cocleft coalgebras and the collection of H-projective module coalgebras over a Hopf algebra H are idempotent radical classes. As applications, we use these idempotent radical classes to give another proofs for a projectivity theorem and a normal basis theorem of Schneider without assuming a bijective antipode.     

Generators for comonoids and universal constructions

We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question in Agore (Proc Am Math Soc 139:855–863, 2011) on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B.     

A synopsis of Combinatorial Integral Geometry

A Lie coalgebra is a coalgebra whose comultiplication Δ : M → M ? M satisfies the Lie conditions. Just as any algebra A whose multiplication ? : A ? A → A is associative gives rise to an associated Lie algebra $\text{L}$(A), so any coalgebra C whose comultiplication Δ : C → C ? C is associative gives rise to an associated Lie coalgebra $\text{L}$^c(C). The assignment C ? $\text{L}$^c(C) is functorial. A universal coenveloping coalgebra Uc(M) is defined for any Lie Lie coalgebra M by asking for a right adjoint U^c to $\text{L}$^c. This is analogous to defining a universal enveloping algebra U(L) for any Lie algebra L by asking for a left adjoint U to the functor $\text{L}$. In the case of Lie algebras, the unit (i.e., front adjunction) 1 → $\text{L}$ o U of the adjoint functor pair U ? $\text{L}$ is always injective. This follows from the Poincaré-Birkhoff-Witt theorem, and is equivalent to it in characteristic zero (x = 0). It is, therefore, natural to inquire about the counit (i.e., back adjunction) $\text{L}$^c o U^c → 1 of the adjoint functor pair $\text{L}$^c ? U^c.Theorem. For any Lie coalgebra M, the natural map$\text{L}$^c(U^cM) → M is surjective if and only if M is locally finite, (i.e., each element of M lies in a finite dimensional sub Lie coalgebra of M).An example is given of a non locally finite Lie coalgebra. The existence of such an example is surprising since any coalgebra C whose diagonal Δ is associative is necessarily locally finite by a result of that theory. The present paper concludes with a development of an analog of the Poincaré-Birkhoff-Witt theorem for Lie algebras which we choose to call the Dual Poincaré-Birkhoff-Witt Theorem and abbreviate by “The Dual PBWθ.” The constraints of the present paper, however, allow only a sketch of this theorem. A complete proof will appear in a subsequent paper. The reader may, however, consult [12], in the meantime, for details. The Dual PBWθ shows for any locally finite Lie coalgebra M the existence (in χ = 0) of a natural isomorphism of the graded Hopf algebras ₀E(U^cM) and ₀E(S^cM) associated to U^cM and to S^cM = U^c(TrivM) when U^c(M) and S^c(M) are given the Lie filtrations. [Just as U^c(M) is the analog of the enveloping algebra U(L) of a Lie algebra L, so S^c(V) is the analog of the symmetric algebra S(V) on a vector space V. Triv(M) denotes the trivial Lie coalgebra structure on the underlying vector space of M obtained by taking the comultiplication to be the zero map.]     

Derivations,automorphisms, and representations of complex ω-Lie algebras

Let (𝔤,ω) be a finite-dimensional non-Lie complex ω-Lie algebra. We study the derivation algebra Der(𝔤) and the automorphism group Aut(𝔤) of (𝔤,ω). We introduce the notions of ω-derivations and ω-automorphisms of (𝔤,ω) which naturally preserve the bilinear form ω. We show that the set Der_ω(𝔤) of all ω-derivations is a Lie subalgebra of Der(𝔤) and the set Aut_ω(𝔤) of all ω-automorphisms is a subgroup of Aut(𝔤). For any three-dimensional and four-dimensional nontrivial ω-Lie algebra 𝔤, we compute Der(𝔤) and Aut(𝔤) explicitly, and study some Lie group properties of Aut(𝔤). We also study representation theory of ω-Lie algebras. We show that all three-dimensional nontrivial ω-Lie algebras are multiplicative, as well as we provide a four-dimensional example of ω-Lie algebra that is not multiplicative. Finally, we show that any irreducible representation of the simple ω-Lie algebra C_α(α≠0,?1) is one-dimensional.     

Chain and distributive coalgebras

We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of Noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive. Infinite dimensional chain coalgebras are finite duals of left Noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D, we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence, cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be seen as tensor products of D^∗, and the divided power coalgebra and can be realized as the generalized path coalgebra of a loop. If D is central, any chain coalgebra is a subcoalgebra of the finite dual of D[[x]].     

A construction for coisotropic subalgebras of Lie bialgebras

Given a Lie bialgebra (g,g^∗), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g^∗. We write down families of examples for the case that g is a classical complex simple Lie algebra.     

Renormalization as a functor on bialgebras

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)⁺), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)⁺), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S¹(B) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)⁺) and the Faà di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra is given. Finally, the bialgebra S(S(B)⁺) is shown to give the same results as the standard renormalization procedure for the scalar field.     

The structure of bialgebra with a Hopf module

Let H be a Hopf algebra, B a bialgebra, and (B, ?, ρ) a right H-Hopf module. Assume that (B, ρ) is a right H-comodule algebra, (B, ?) is a right H-module coalgebra, and let A = B ^{co H} = {a ∈ B | ρ(a) = a ? 1}. Then we prove that B has a factorization of A□ _ρ ^? (the underlying space is A ? H) as a bialgebra, which generalizes Radford’s factorization of bialgebras with projection [12].     

<Emphasis Type="Italic">n</Emphasis>-Lie Structures That Are Generated by Wronskians

We study the (k + 1)-Lie structures, k-left commutative and homotopy (k + 1)-Lie structures with multiplication generated by Wronskians and prove that the nontrivial structures of n-Lie algebras appear only in the case of small characteristic.Original Russian Text Copyright © 2005 Dzhumadil’daev A. S.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 759–773, July–August, 2005.     

On the combinatorial rank of a graded braided bialgebra

Let B be a graded braided bialgebra. Let S(B) denote the algebra obtained dividing out B by the two sided ideal generated by homogeneous primitive elements in B of degree at least two. We prove that S(B) is indeed a graded braided bialgebra quotient of B. It is then natural to compute S(S(B)), S(S(S(B))) and so on. This process yields a direct system whose direct limit comes out to be a graded braided bialgebra which is strongly N-graded as a coalgebra. Following V.K. Kharchenko, if the direct system is stationary exactly after n steps, we say that B has combinatorial rank n and we write κ(B)=n. We investigate conditions guaranteeing that κ(B) is finite. In particular, we focus on the case when B is the braided tensor algebra T(V,c) associated to a braided vector space (V,c), providing meaningful examples such that κ(T(V,c))≤1.     

Additive Lie (ξ-Lie) Derivations and Generalized Lie (ξ-Lie) Derivations on Prime Algebras

The additive(generalized)ξ-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumptions, that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive(generalized)ξ-Lie derivation with ξ = 1 if and only if it is an additive(generalized)derivation satisfying L(ξA)= ξL(A)for all A. These results are then used to characterize additive(generalized)ξ-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.     

Laguerre Polynomials, Restriction Principle, and Holomorphic Representations of SL(2,R)

The restriction principle is used to implement a realization of the holomorphic representations of SL(2,R) on L ² (R ⁺,t dt) by way of the standard upper half plane realization. The resulting unitary equivalence establishes a correspondence between functions that transform according to the character e^–i(2n++1); under rotations and the Laguerre polynomials. The standard recursion relations amongst Laguerre polynomials are derived from the action of the Lie algebra.     

Lie-Rinehart bialgebras for crossed products

Let g be a Lie algebra over a field K. Endow the polynomial ring K[t] with a g-action by derivations and consider the resulting crossed product K[t]⊙_θg in the category of (K,K[t])-Lie algebras. We explore Lie-Rinehart bialgebra structures on (K[t],K[t]⊙_θg) that arise via compatible pairs and ε-dynamical r-matrices. We give a complete classification for .