Contractions of Low-Dimensional Nilpotent Jordan Algebras

In this article, we classify the laws of three-dimensional and four-dimensional nilpotent Jordan algebras over the field of complex numbers. We describe the irreducible components of their algebraic varieties and extend contractions and deformations among their isomorphism classes. In particular, we prove that 𝒥² and 𝒥³ are irreducible and that 𝒥⁴ is the union of the Zariski closures of the orbits of two rigid Jordan algebras.     

Differential algebras and simple Jordan superalgebras

In [14], a new example is constructed of a unital simple special Jordan superalgebra J over the field of reals. It turns out that J is a subsuperalgebra of a Jordan superalgebra of vector type but it cannot be isomorphic to a superalgebra of such a type. Moreover, the superalgebra of fractions of J is isomorphic to a Jordan superalgebra of vector type. In the present article, we find a similar example of a Jordan superalgebra. It is constructed over a field of characteristic 0 in which the equation t ² + 1 = 0 has no solutions.     

Strict proper nilness modulo an absorber

An element z of a Jordan system J is strictly properly nilpotent if it is nilpotent in all homotopes of all extensions of J; it is strictly properly nilpotent of bounded index if there is a bound on its indices of nilpotence in all these extensions. We showed in a previous paper that the strictly properly nilpotent elements (resp. those of bounded index) form an ideal, by exhibiting that ideal as an Amitsur shrinkage. In this paper we prove by combinatorial methods (the exponential law for power series and Jordan Binomial Theorems) a modular generalization of this: the elements strictly properly nilpotent (resp. boundedly so) modulo the absorber Q of an inner ideal K form an ideal. As a corollary we obtain Zelmanov’s Nilness-mod-the-Absorber-Theorem that the ideal generated by Q is nil mod K. From this we re-derive his Primitive Absorber and Exceptional Heart Theorems (improved to show the heart is S(J), not just S( (J)³), two results crucial to the structure of primitive Jordan systems, using a triple-system version of Zelmanov's KKT-specialization for inner ideals in a Jordan pair.     

The Trace in Banach Jordan Pairs

The algebraic trace form (as defined by O. Loos) of an element(x, y) of a (complex) Banach Jordan pair V, where x or y isin the socle, is equal to the sum of the products of all spectralvalues and their multiplicity. The trace form is calculatedfor two examples, the Banach Jordan pair of bounded linear operatorsbetween two Banach spaces, and the Banach Jordan pair of a quadraticform. Using analytic multifunctions, it is also shown that thecomplement of the socle of a Banach Jordan pair V is eitherdense or empty. In the last case, V has finite capacity. 1991Mathematics Subject Classification 17C65, 46H70.     

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.     

On the socle of a noncommutative Jordan algebra

In this paper we study some questions related to the socle of a nondegenerate noncommutative Jordan algebra. First we show that elements of finite rank belong to the socle, and that every element in the socle is von Neumann regular and has finite spectrum. Next we show that for Jordan Banach algebras the socle coincides with the maximal von Neumann regular ideal. For a nondegenerate noncommutative Jordan algebra, the annihilator of its socle can be regarded as a radical which is, generally, larger than Jacobson radical. Moreover, a nondegenerate noncommutative Jordan algebra whose socle has zero annihilator is isomorphic to a subdirect sum of primitive algebras having nonzero socle (which were described in [4]). Finally, these results are specialized to the particular case of an alternative algebra.The authors wish to thank the referee for his suggestions for improving the presentation of the paper.     

Formations generated by a group of socle length 2

Gaschütz conjectured that a formation generated by a finite group contains only finitely many subformations. In the present article we prove this conjecture for the groups of socle length at most 2. (We say that a group has socle length 1 if it coincides with its socle and has socle length 2 if its quotient by the socle has socle length 1.) Earlier Gaschütz’s conjecture was proven in several particular cases including all soluble groups.     

Continuity of generalized derivations on JB*-algebras

We prove that every generalized Jordan derivation D from a JB?-algebra 𝒜 into itself or into its dual space is automatically continuous. In particular, we establish that every generalized Jordan derivation from a C?-algebra to a Jordan Banach module is continuous. As a consequence, every generalized derivation from a C?-algebra to a Banach bimodule is continuous.     

Graded automorphism group of TKK algebra

The classification of extended affine Lie algebras of type A_1 depends on the Tits-Kantor- Koecher (TKK) algebras constructed from semilattices of Euclidean spaces.One can define a unitary Jordan algebra J(S) from a semilattice S of R~v (v≥1),and then construct an extended affine Lie algebra of type A_1 from the TKK algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction.In R~2 there are only two non-similar semilattices S and S′,where S is a lattice and S′is a non-lattice semilattice.In this paper we study the Z~2-graded automorphisms of the TKK algebra T(J(S)).     

Maps preserving product X*Y + YX* on factor von Neumann algebras

Let 𝒜 and ? be two factor von Neumann algebras. In this article, we prove that a nonlinear bijective map Φ?:?𝒜?→?? satisfies Φ(X*?Y?+?YX*)?=Φ(X)*Φ(Y)?+?Φ(Y)Φ(X)* (?X,?Y?∈?𝒜), if and only if Φ is a *-ring isomorphism. In particular, if 𝒜 and ? are type I factors, then Φ is a unitary isomorphism or conjugate unitary isomorphism.     

Artinian Level Algebras of Low Socle Degree

In this article we study Hilbert functions and isomorphism classes of Artinian level local algebras via Macaulay's inverse system. Upper and lower bounds concerning numerical functions admissible for level algebras of fixed type and socle degree are known. For each value in this range we exhibit a level local algebra with that Hilbert function, provided that the socle degree is at most three. Furthermore, we prove that level local algebras of socle degree three and maximal Hilbert function are graded. In the graded case, the extremal strata have been parametrized by Cho and Iarrobino.     

Indecomposable Representations of the Superalgebra B(1,2)

Finite-dimensional indecomposable superbimodules over the superalgebra B(1,2) are treated. We propound a method for constructing indecomposable alternative superbimodules over B(1,2) containing a given socle (such can be presented by any irreducible module over B(1,2)). The method is based on adding on the Jordan basis. Also, for the characteristic 3 case, we give examples of Jordan indecomposable superbimodules which are not alternative.     

The Lie algebra of skew-symmetric elements and its application in the theory of Jordan algebras

We prove that the Lie algebra of skew-symmetric elements of the free associative algebra of rank 2 with respect to the standard involution is generated as a module by the elements [a, b] and [a, b]³, where a and b are Jordan polynomials. Using this result we prove that the Lie algebra of Jordan derivations of the free Jordan algebra of rank 2 is generated as a characteristic F-module by two derivations. We show that the Jordan commutator s-identities follow from the Glennie-Shestakov s-identity.     

Isomorphism Classes of Certain Artinian Gorenstein Algebras

In this paper we attack the problem of the classification, up to analytic isomorphism, of Artinian Gorenstein local k-algebras with a given Hilbert Function. We solve the problem in the case the square of the maximal ideal is minimally generated by two elements and the socle degree is high enough.     

The isomorphism relation between tree-automatic Structures

An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is neither a Σ₂¹-set nor a Π₂¹-set.     

Penalization methods for reflecting stochastic differential equations with jumps

The problem of approximation of a solution to a reflecting stochastic differential equation (SDE) with jumps by a sequence of solutions to SDEs with penalization terms is considered. The approximating sequence is not relatively compact in the Skorokhod topology J ¹ and so the methods of approximation based on the J ¹-topology break down. In the paper, we prove our convergence results in the S-topology on the Skorokhod space D(R⁺,?R ^d ) introduced recently by Jakubowski. The S-topology is weaker than J ¹ but stronger than the Meyer-Zheng topology and shares many useful properties with J ¹.     

Coefficient Ideal of Ideals Generated by Monomials

In a commutative Noetherian ring R, the coefficient ideal of I relative to J is the largest ideal 𝔟 for which I𝔟 =J𝔟 when I is integral over J. In this article, we will give a simple algorithm to compute 𝔞(I, J) when I, J are ideals in a polynomial ring R = k[X ₁,…, X _d ] generated by monomials and J is a parameter ideal. We use the concept of socle sequence. Also we will show that the reduction number r _J (I) is also computed by our algorithm.     

Quotient polytopes of cyclic polytopes part II: Stability of thef-vector and of thek-skeleton

This paper is the continuation of an earlier paper on quotient polytopesC(v, 2m)/F of cyclic polytopes and the associated quotient complexesC(V, 2m)/J. Here, we study mainly what changes in the faceJ do not affect thef-vector of the quotientC(V, 2m)/J. In the last section we examine the corresponding question fork-skeleta, i.e., what changes inJ do not affect the isomorphism type of skel _k C(V, 2m)/J.     

Homological stability for the mapping class groups of non-orientable surfaces

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum built from the canonical bundle over the Grassmannians of 2-planes in ℝ ⁿ⁺². In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i – this is the non-oriented analogue of the Mumford conjecture.