Abstract :
I. Kaplansky discovered a simple 3-dimensional half-unital Jordan superalgebra κ3(Φ) spanned over a field Φ by e0, ξ1, η1 with e0·ξ1 = imageξ1, e0·η1 = imageη1, image1·ξ = e0. This Tiny Kaplansky superalgebra is special with specialization [formula] in the 2 × 2 matrices M2(image) over the algebra image = Φ left angle bracketLt, d/dt] of differential operators on the polynomial algebra Φ[t]. In this paper we obtain a general class of half-unital Kaplansky superalgebras κ(image) = κ0(A) circled plus κ1(M) with κ0(a)·κ0(b) = κ0(ab), κ0(a)·κ1(m) = imageκ1(am), κ1(m)·κ1(n) = κ0(m × n) built from a bracket module image = (A, M, ×) consisting of a scalar algebra A, a unital A-module M, and a skew product on M × M to A. These superalgebras are simple iff image is simple as bracket module iff A contains no proper M-invariant ideals (ideals invariant under all odd derivations Dm,n(a) = (am) × n − a(m × n)) and M × M = A. κ(image) is Jordan iff image satisfies a Jacobi identity ∑cyclic (mi × mj)mk = 0 and a derivation identity 2a(m × n) = am × n + m × an. The two basic examples of bracket modules are nested in scalar algebras F: A = F0, M = F1 for F0 subset of F1 subset of F with F0 a unital subalgebra of F and F1 an F0-module with bracket product ƒ × g = D(ƒ) g − ƒD(g) for D: F1 → F0 a derivation of F0-modules. When D is a global derivation of F the Jordan superalgebra κ(F0, F1, ×D) is special with specialization [formula] in the 2 × 2 matrices image2(image) over the algebra image = Φleft angle bracketLF1, Dright-pointing angle bracket on F. The first basic example is the Tiny Kaplansky module image3(Φ) (A = Φ1, M = Φ1 circled plus Φt, F = Φ[t], D = d/dt, κ(image3(Φ)) = κ3(Φ)), which is simple iff Φ is a field. The second basic example is the full derivation module image(F, D) (A = M = F, ƒ × g = D(ƒ ) g − ƒD(g) for a derivation D of F), which is simple iff F is D-simple, e.g., F the infinite-dimensional Φ[t] or Φ[[t]] for D = d/dt over a field of characteristic 0, or F the p-dimensional truncated polynomial algebra Φ[tp], D = d/dtp over a field of characteristic p. Our main result is that if image is simple of characteristic p > 0 (or is simple of characteristic 0 with A local, e.g., algebraic over a field), then image is either tiny or full; if image is simple of characteristic 0 then image imbeds in a full image(F, D) over a field F. In particular, all simple Kaplansky superalgebras κ(image) are special.
