Isomorphic Lie Research Articles

Twisted cocycles of lie algebras and corresponding invariant functions

We consider finite-dimensional complex Lie algebras. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie algebras. A special case is generalization of 1-cocycles with respect to the adjoint representation – so called (α,β,γ)-derivations. Parametric sets of spaces of cocycles allow us to define complex functions which are invariant under Lie isomorphisms. Such complex functions thus represent useful invariants – we show how they classify three and four-dimensional Lie algebras as well as how they apply to some eight-dimensional one-parametric nilpotent continua of Lie algebras. These functions also provide necessary criteria for existence of 1-parametric continuous contraction.

Multiplicative ∗-Lie isomorphisms between factors

Let M , N be factors. If Φ : M → N is a multiplicative ∗-Lie isomorphism, then Φ is of the form σ + τ , where σ is a linear ∗-isomorphism, or a conjugate linear ∗-isomorphism, or the negative of a linear ∗-anti-isomorphism, or the negative of a conjugate linear ∗-anti-isomorphism of M onto N and τ is a mapping of M into C I = { λ I | λ ∈ C } which maps commutators into zero.

Multiplicative Lie Isomorphisms Between Prime Rings

Let ℛ be a prime ring with 1 containing a nontrivial idempotent E, and let ℛ′ be another prime ring. If Φ:ℛ → ℛ′ is a multiplicative Lie isomorphism, then Φ(T + S) = Φ(T) + Φ(S) + Z′ T,S for all T, S ∈ ℛ, where Z′ T,S is an element in the center 𝒵′ of ℛ′ depending on T and S.

On [formula omitted]-derivations of Lie algebras and corresponding invariant functions

We consider finite dimensional complex Lie algebras. We generalize the concept of Lie derivations via certain complex parameters and obtain various Lie and Jordan operator algebras as well as two one-parametric sets of linear operators. Using these parametric sets, we introduce complex functions with a fundamental property — invariance under Lie isomorphisms. One of these basis-independent functions represents a complete set of invariant(s) for three-dimensional Lie algebras. We present also its application to physically motivated examples in dimension 8.

Lie isomorphisms of reflexive algebras

A Lie isomorphism ϕ between algebras is called trivial if ϕ = ψ + τ , where ψ is an (algebraic) isomorphism or a negative of an (algebraic) anti-isomorphism, and τ is a linear map with image in the center vanishing on each commutator. In this paper, we investigate the conditions for the triviality of Lie isomorphisms from reflexive algebras with completely distributive and commutative lattices (CDCSL). In particular, we prove that a Lie isomorphism between irreducible CDCSL algebras is trivial if and only if it preserves I-idempotent operators (the sum of an idempotent and a scalar multiple of the identity) in both directions. We also prove the triviality of each Lie isomorphism from a CDCSL algebra onto a CSL algebra which has a comparable invariant projection with rank and corank not one. Some examples of Lie isomorphisms are presented to show the sharpness of the conditions.

Commuting traces and commutativity preserving maps on triangular algebras

Let A be a triangular algebra. The problem of describing the form of a bilinear map B : A × A → A satisfying B ( x , x ) x = x B ( x , x ) for all x ∈ A is considered. As an application, commutativity preserving maps and Lie isomorphisms of certain triangular algebras (e.g., upper triangular matrix algebras and nest algebras) are determined.

Commuting Traces of Biadditive Maps Revisited

Let R be a prime ring with char (R) ≠ 2. It is known that the form of a biadditive map B : R × R → R satisfying [B(x, x), x] = 0 for all x ∈ R can be described provided that R does not satisfy the standard polynomial identity S 4. In this note we prove a linear algebraic result from which we are able to derive that the description of B is the same even when R satisfies S 4. This makes it possible to complete the results on the structure of Lie isomorphisms and Lie derivations on prime rings.

Lie Isomorphisms of Prime Rings Satisfying St4

Let α : RR’ be a Lie isomorphism of prime rings. Suppose that if R satisfies the standard polynomial identity St4 of degree 4. We prove that α has a decomposition into the sum of a homomorphism (or the negative of an anithomomorphism) plus an additive map into the center of R’.

On Herstein's Lie Map Conjectures, III

In the two papers, “On Herstein's Lie Map Conjectures, I and II,” solutions of all of Herstein's problems concerning Lie isomorphisms and derivations were given under the assumption that the algebras involved were of “sufficiently high” dimension. In the present paper we remove this restriction, thereby solving Herstein's problems in full generality.

LIE ISOMORPHISMS IN *-PRIME GPI RINGS WITH INVOLUTION

Let $R$ and $S$ be *-prime GPI rings with involution, with respective skew elements $K$ and $L$, with respective extended centroids $C$ and $D$, and let $\alpha :[K, K]/[K, K]\cap C\to [L, L]\cap D$ be a Lie isomorphism. If both involutions are of the second kind it is shown that $\alpha$ is determined by a related associative isomorphism and if the involutions are of different kinds it is shown that such a map $\alpha$ cannot exist (modulo some low-dimensional counterexamples).

Time-dependent Schrödinger equations having isomorphic symmetry algebras. II. Symmetry algebras, coherent and squeezed states

Using the transformations from paper I, we show that the Schrödinger equations for (1) systems described by quadratic Hamiltonians, (2) systems with time-varying mass, and (3) time-dependent oscillators all have isomorphic Lie space–time symmetry algebras. The generators of the symmetry algebras are obtained explicitly for each case and sets of number-operator states are constructed. The algebras and the states are used to compute displacement-operator coherent and squeezed states. Some properties of the coherent and squeezed states are calculated. The classical motion of these states is demonstrated.

Nilmanifolds and associated Lie algebras over the integers

We consider the correspondence between nilmanifolds and Lie algebras with rational basis, and we dene spectral sequences converging to the respective cohomologies. The E2 terms of the spectral sequences are the cohomolgies of isomorphic graded Lie algebras. Each nilmanifold gives rise to a Lie algebra with rational basis. We give an example which illustrates that not all such Lie algebras correspond to nilmanifolds. Given a Lie algebra with rational basis we give a construction that produces a nilmanifold with Lie algebra that is rationally equivalent to the starting Lie algebra.

Lie Isomorphisms of Nest Algebras

In this paper we characterize linear mapsϕbetween two nest algebras T(N) and T(M) which satisfy the property thatϕ(AB−BA)=ϕ(A)ϕ(B)−ϕ(B)ϕ(A) for allA, B∈T(N). In particular, it is shown that such isomorphisms only exist if N is similar to M or M⊥.

On lie isomorphisms in prime rings with involution

We describe Lie isomorphisms of skew elements of prime rings of characteristic not 2 with involution of the first kind thus extending the corresponding result of Beidar, Martindale and Mikhalev to the case of characteristic 3.

Continuity of Lie Isomorphisms of Banach Algebras

We prove that if A and B are semisimple Banach algebras, then the separating subspace of every Lie isomorphism from A onto B is contained in the centre of B. 1991 Mathematics Subject Classification 17B40, 17B60, 46H40.

Continuity of Lie mappings of the skew elements of Banach algebras with involution

Let A A and B B be centrally closed prime complex Banach algebras with linear involution. If A A is semisimple, then any Lie derivation of the skew elements of A A is continuous and any Lie isomorphism from the skew elements of B B onto the skew elements of A A is continuous.

Displacement-operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras

In this paper we use the Lie algebra of space-time symmetries to construct states which are solutions to the time-dependent Schrödinger equation for systems with potentials V(x,τ)=g(2)(τ)x2+g(1)(τ)x+g(0)(τ). We describe a set of number-operator eigenstates states, {Ψn(x,τ)}, that form a complete set of states but which, however, are usually not energy eigenstates. From the extremal state, Ψ0, and a displacement squeeze operator derived using the Lie symmetries, we construct squeezed states and compute expectation values for position and momentum as a function of time, τ. We prove a general expression for the uncertainty relation for position and momentum in terms of the squeezing parameters. Specific examples, all corresponding to choices of V(x,τ) and having isomorphic Lie algebras, will be dealt with in the following paper (II).

Group-Theoretic analysis of the mixing angle in the electroweak gauge group

In this paper the authors provide strong mathematical support for the idea that the experimentally measured magnitude 1 - M{sub W}{sup 2}/M{sub Z}{sup 2} associated with sin{sup 2}{theta}{sub w} in the standard model of electroweak interactions cannot be simultaneously identified with the squared quotient of the electric charge by the SU(2) charge, e{sup 2}/g{sup 2}. In fact, the natural, mathematical requirement that the Weinberg rotation between the gauge fields associated with the third component of the {open_quotes}weak isospin{close_quotes} (T{sub 3}) and the hypercharge (Y) proceeds from a global Lie-group homomorphism of the SU(2) {circle_times} U(1){sub y} gauge group in some locally isomorphic Lie group [which then proves to be U(2)], and not from a local (Lie algebra) isomorphism, imposes strong restrictions so as to fix the single value e{sup 2}/g{sup 2} = 1/2. The two definitions of sin{sup 2}{theta}{sub w} can only be identified in the asymptotic limit corresponding to an earlier stage of the universe before the spontaneous symmetry breaking had taken place.

Lie Isomorphisms in Prime Rings with Involution

Let R and R′ be prime rings with involutions of the first kind and with respective Lie subrings of skew elements K and K′. Furthermore assume ( RC : C) ≠ 1, 4, 9, 16, 25, 64, where C is the extended centroid of R. It is shown that any Lie isomorphism of K onto K′ can be extended uniquely to an associative isomorphism of 〈 K〉 onto 〈 K′〉, where 〈 K〉 and 〈 K′〉 are respectively the associative subrings generated by K and K′.

Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings

Biadditive mappings B: R × R → R where R is a prime ring with certain additional properties, satisfying B(x, x)x = xB(x, x) for all x ∈ R, are characterized. As an application we determine the structures of commutativity-preserving mappings, Lie isomorphisms, and Lie derivations of certain prime rings