Isomorphic Lie Research Articles

Infinitely many versions of SU(∞)

We give an explicit example of - infinitely many - pairwise non isomorphic Lie algebras L Λ (Λ irrational) each of which is a limit of su( N) as N → ∞, in the sense that for each Λ there exists a sequence N i ( Λ) of natural numbers N 1 < N 2 < … < N i < … and basis T a ( N i ) of su( N i ), such that the corresponding structure constants ƒ abc(N i) converge to those of L Λ fo N i → ∞.

On when a separable field extension in characteristic p > 0 is determined by its Frobenius structure, I

Let G c Aut,( V) be a subgroup of automorphisms of a finite-dimensional vector space V over a field k. A vector u E I’ is said to be a G-orbit basis generator for V if the set G(u) is a linear basis for I’. In this paper we continue the study begun in [Rl, R2] of the following question: If U, U are G-orbit basis generators for V then when are the stabilizers G, and G, conjugate? The conjugacy question is a more general formulation of a question concerning finite-dimensional separable field extensions E, F over an infinite field k of characteristic p > 0. Let k be a field of characteristic p > 0 and suppose E and F are n-dimensional separable field extensions of k. Recall that the associative structure of any commutative algebra A over k affords A a restricted Lie (Frobenius) structure Y(A) by [ab] =0 and aCpl =up for a, h E A. A natural question to ask about the pair E and F is the following: If Z(E) and Z(F) are isomorphic Lie algebras, then when are E and F isomorpic field extensions? Suppose P(E) N P(F). It does not necessarily follow that E = F;

Lie isomorphisms of the skew elements of a prime ring with involution

Lie isomorphisms of the skew elements of a prime ring with involution

Lie isomorphisms of the skew elements of a simple ring with involution

Lie isomorphisms of the skew elements of a simple ring with involution

Lie isomorphisms of factors

Lie isomorphisms of factors

Lie Isomorphisms of Factors

Lie Isomorphisms of Factors

Lattice isomorphic solvable Lie algebras

Let L be a Lie algebra over a field κ of any characteristic, and consider the lattice ℒ(L) of all subalgebras of L. In this paper we prove that if L and M are lattice isomorphic Lie algebras, over a field of any characteristic, and L′ and M′ are nilpotent, then the difference between the orders of solvability of L and M differs by at most one.

Lie Isomorphisms of Derived Rings of Simple Rings

Lie Isomorphisms of Derived Rings of Simple Rings

Lie Isomorphisms of Prime Rings

Lie Isomorphisms of Prime Rings

Lie isomorphisms of derived rings of simple rings

Lie isomorphisms of derived rings of simple rings

Lie isomorphisms of prime rings

for all x, y E S. Our interest and viewpoint toward the study of Lie isomorphisms of rings was originally (and still is) inspired by the work done by I. N. Herstein on generalizing classical theorems on the Lie structure of total matrix rings to results on the Lie structure of arbitrary simple rings. In our case the starting point was the realization that it should be possible to extend the following theorem of L. Hua [1]: every Lie automorphism of the ring R of all n x n matrices over a division ring, n >2, characteristic & 2, 3, is of the form a + -r, where a is either an automorphism or the negative of an antiautomorphism of R and X is an additive mapping of R into its center which maps commutators into zero. Indeed, using some of Hua's techniques and some valuable suggestions due to Nathan Jacobson, we were in [2], roughly speaking, able to obtain the same conclusion under the weaker assumptiop that R was merely a primitive ring possessing three orthogonal idempotents whose sum was 1. In a recent paper [3], while making the stronger assumption that R was simple, we were able to lower the number of idempotents from three to two. For the most part, the same techniques were used in this second paper, although a tensor product method due to Jacobson was to replace tedious calculations involving matrix units due to Hua, and some results of Herstein on Lie ideals of simple rings seemed necessary. Our goal in this paper is Theorem 11, in which we extend the above results to the situation where R is a prime ring with two orthogonal idempotents whose sum is 1. Primeness is a natural generalization of simplicity and primitivity, and, in the sense of keeping free of the radical and of (sub)direct sums of ideals, it is perhaps the strongest generalization one may make. Whether the assumption of idempotents is necessary or not is still a major open question. In all our work on the subject (including the present paper) our arguments rest heavily on the presence of a nontrivial idempotent. A successful removal of the assumption of idempotents would certainly require totally new methods; one would, for example, have to face the situation of an arbitrary division ring.

Lie Isomorphisms of Primitive Rings

Lie Isomorphisms of Primitive Rings

Lie isomorphisms of primitive rings

for all x, yER. In this paper we study Lie isomorphisms of a primitive ring R onto a primitive ring R', where we assume that the characteristic of R is different from 2 and 3 and that R contains three nonzero orthogonal idempotents whose sum is the identity. Such isomorphisms will be shown to be of the form o+r, where ois either an isomorphism or the negative of an anti-isomorphism of R into a primitive ring L' containing R' and r is an additive mapping of R into the center of L' which maps commutators into zero. This generalizes a theorem of Hua [1], who obtained the above result in the case that R (= '= L') was the ring of all n X n matrices over a division ring, n> 2. On the other hand, due to our requirement concerning idempotents, we fall far short of providing a general solution to Herstein's conjecture [2] that the result holds for arbitrary simple rings. An important part of our proof consists of a repetition of arguments involving matrix units used by Hua [1], and for the sake of completeness (and also because of the relative inaccessability of Hua's paper) we shall reproduce his proofs in some detail when the occasion demands. The author is especially grateful to Professor Nathan Jacobson for several valuable comments towards improving an earlier version of this paper. In the case of simple rings he indicated how all calculations involving matrix units can be eliminated, and he suggested extending our original result to the more general case of primitive rings.

The Lie algebra of a smooth manifold

It is well known that certain topological spaces are determined by rings of continuous real functions defined over them [1; 2; 3],1 and for differentiable manifolds the functions may be differen tiable [4; 7]. In this note we prove that the Lie algebra of all tangent vector fields with compact supports on an infinitely differentiable manifold determines the manifold, and that two such manifolds with isomorphic Lie algebras are differentiably homeomorphic. This Lie algebra has been studied by H. Cartan [6], and for the case of analytic manifolds by Chevalley [5]. Let X be an infinitely differentiable (smooth) manifold and let R be the real numbers. Denote by D the algebra over R of all infinitely differentiable functions on X with the natural addition and multiplication, and by Do the subalgebra of D consisting of those functions with compact supports (that is, which vanish outside of compact sets). A tangent vector field L is a linear operator on D to D which is also an abstract derivative, that is, if a, bER and f, gED, then L(af +bg) =aL(f) +bL(g) and L(fg) =fL(g) +gL(f). The set et of all tangent vector fields, with the obvious addition and multiplication by real numbers, becomes a Lie algebra2 over R when the product of L1 and L2 is defined by