On absolute valued algebras with involution

Let A be an absolute valued algebra with involution, in the sense of Urbanik [K. Urbanik, Absolute valued algebras with an involution, Fund. Math. 49 (1961) 247-258]. We prove that A is finite-dimensional if and only if the algebra obtained by symmetrizing the product of A is simple, if and only if eA_s = A_s, where e denotes the unique nonzero self-adjoint idempotent of A, and A_s stands for the set of all skew elements of A. We determine the idempotents of A, and show that A is the linear hull of the set of its idempotents if and only if A is equal to either McClay’s algebra [A.A. Albert, A note of correction, Bull. Amer. Math. Soc. 55 (1949) 1191], the para-quaternion algebra, or the para-octonion algebra. We also prove that, if A is infinite-dimensional, then it can be enlarged to an absolute valued algebra with involution having a nonzero idempotent different from the unique nonzero self-adjoint idempotent.     

A chain of evolution algebras

We introduce a notion of chain of evolution algebras. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies an analogue of Chapman-Kolmogorov equation. We give several examples (time homogenous, time non-homogenous, periodic, etc.) of such chains. For a periodic chain of evolution algebras we construct a continuum set of non-isomorphic evolution algebras and show that the corresponding discrete time chain of evolution algebras is dense in the set. We obtain a criteria for an evolution algebra to be baric and give a concept of a property transition. For several chains of evolution algebras we describe the behavior of the baric property depending on the time. For a chain of evolution algebras given by the matrix of a two-state evolution we define a baric property controller function and under some conditions on this controller we prove that the chain is not baric almost surely (with respect to Lebesgue measure). We also construct examples of the almost surely baric chains of evolution algebras. We show that there are chains of evolution algebras such that if it has a unique (resp. infinitely many) absolute nilpotent element at a fixed time, then it has unique (resp. infinitely many) absolute nilpotent element any time; also there are chains of evolution algebras which have not such property. For an example of two dimensional chain of evolution algebras we give the full set of idempotent elements and show that for some values of parameters the number of idempotent elements does not depend on time, but for other values of parameters there is a critical time t_c such that the chain has only two idempotent elements if time t?t_c and it has four idempotent elements if time t<t_c.     

Invariance of dimension of p-subspaces in bernstein algebras *

The purpose of this paper is to prove that some vector subspaces, called p-subspaces, obtained from the Peirce decomposition of a Bernstein algebra A relative to an idempotent have dimensions which are independent of the idempotent used to decompose A. In particular, for Bernstein-Jordan algebras, this fact is true for every such subspace and this implies that all p-subspaces of a Bernstein algebra, contained in V, for A = Ke + U + V, have invariant dimension. Finally we classify all p-subspaces of degree ≥ 3, contained in U, in a Bernstein algebra A, relative to the invariance (or not) of dimension.     

Star partial order on B(H)

Let H be an infinite-dimensional complex Hilbert space and let B(H) be the algebra of all bounded linear operators on (H). In the paper the equivalent definition of the star partial order on B(H), using selfadjoint idempotent operators, is introduced. Also some properties of the generalized concept of order relations on B(H), defined with the help of idempotent operators, are investigated.     

On the Leibniz (co)homology of the Lie algebra of the Euclidean group

The Lie algebra of the Euclidean group is an abelian extension of the orthogonal Lie algebra. We compute its Leibniz (co)homology. It is computed via the identification of certain orthogonal invariants and shown to be an algebra generated by a n−1-fold tensor and an n-fold tensor.     

Finitely Related Clones and Algebras with Cube Terms

Aichinger et al. (2011) have proved that every finite algebra with a cube-term (equivalently, with a parallelogram-term; equivalently, having few subpowers) is finitely related. Thus finite algebras with cube terms are inherently finitely related??every expansion of the algebra by adding more operations is finitely related. In this paper, we show that conversely, if A is a finite idempotent algebra and every idempotent expansion of A is finitely related, then A has a cube-term. We present further characterizations of the class of finite idempotent algebras having cube-terms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomial-time algorithm for determining if the algebra has a cube-term. We also determine the maximal non-finitely related idempotent clones over A. The number of these clones is finite.     

The generalized entropic property for a pair of operations

In this paper a generalized entropic property is defined for a pair of operations. We show that for an idempotent algebra A = (A, f, g) with two ternary operations, if one of f or g is commutative and the pair of operations (f, g) satisfies the generalized entropic property, then (f, g) is entropic. Also, it is proved that every idempotent, commutative algebra A = (A, f, g) with a ternary and a binary operation, satisfying the generalized entropic property, is entropic.     

Algebras whose right nucleus is a central simple algebra

We generalize Amitsur's construction of central simple algebras over a field F which are split by field extensions possessing a derivation with field of constants F to nonassociative algebras: for every central division algebra D over a field F of characteristic zero there exists an infinite-dimensional unital nonassociative algebra whose right nucleus is D and whose left and middle nucleus are a field extension K of F splitting D, where F is algebraically closed in K.We then give a short direct proof that every p-algebra of degree m, which has a purely inseparable splitting field K of degree m and exponent one, is a differential extension of K and cyclic. We obtain finite-dimensional division algebras over a field F of characteristic $p>0$ whose right nucleus is a division p-algebra.     

Preunits and weak crossed products

Using the notion of a preunit and the properties of idempotent morphisms, we give a general notion of a crossed product of an algebra A and an object V both living in a monoidal category C. We endow A⊗V with a multiplication and an idempotent morphism, whose image inherits the multiplication. Sufficient conditions for these multiplications to be associative are given. If the product on A⊗V has a preunit, the related idempotent is given in terms of the preunit, and its image has an algebra structure. A characterization of crossed products with preunit is given, and it is used to recover classical examples of crossed products and to study crossed products in weak contexts. Finally crossed products of an algebra by a weak bialgebra are recovered using this theory.     

A direct product decomposition of QMV algebras

We study the direct product decomposition of quantum many-valued algebras (QMV algebras) which generalizes the decomposition theorem of ortholattices (orthomodular lattices).In detail,for an idempo- tent element of a given QMV algebra,if it commutes with every element of the QMV algebra,it can induce a direct product decomposition of the QMV algebra.At the same time,we introduce the commutant C(S) of a set S in a QMV algebra,and prove that when S consists of idempotent elements,C(S) is a subalgebra of the QMV algebra.This also generalizes the cases of orthomodular lattices.     

Maps preserving the idempotency of products of operators

Let B(X) be the algebra of all bounded linear operators on a complex Banach space X. We give the concrete form of every unital surjective map φ on B(X) such that AB is a non-zero idempotent if and only if φ(A)φ(B) is for all A,B∈B(X) when the dimension of X is at least 3.     

Trilinear products and comtrans algebra representations

Comtrans algebras are modules over a commutative ring R equipped with two trilinear operations: a left alternative commutator and a translator satisfying the Jacobi identity, the commutator and translator being connected by the so-called comtrans identity. The standard construction of a comtrans algebra uses the ternary commutator and translator of a trilinear product. If 6 is invertible in R, then each comtrans algebra arises in this standard way from the so-called bogus product.Consider a vector space E of dimension n over a field R. While the dimension of the space of all trilinear products on E is n⁴, the dimension of the space of all comtrans algebras on E is less, namely . The paper determines which trilinear products may be represented as linear combinations of the commutator and translator of a comtrans algebra. For R not of characteristic 3, the necessary and sufficient condition for such a representation is the strong alternativityxxy+xyx+yxx=0 of the trilinear product xyz. For R also not of characteristic 2, it is shown that the representation may be given by the bogus product. A suitable representation for the characteristic 2 case is also obtained.     

η-Simple semigroups without zero and η ∗-simple semigroups with a least non-zero idempotent

A semigroup S is called η-simple if S has no semilattice congruences except S×S. Tamura in (Semigroup Forum 24:77–82, 1982) studied η-simple semigroups with a unique idempotent. In the present paper we consider a more general situation, that is, we investigate η-simple semigroups (without zero) with a least idempotent. Moreover, we study η ^?-simple semigroups with zero which contain a least non-zero idempotent.     

Products of idempotent matrices over integral domains

We say that a ring R has the idempotent matrices property if every square singular matrix over R is a product of idempotent matrices. It is known that every field, and more generally, every Euclidean domain has the idempotent matrices property. In this paper we show that not every integral domain has the idempotent matrices property and that if a projective free ring has the idempotent matrices property then it must be a Bezout domain. We also show that a principal ideal domain has the idempotent matrices property if and only if every fraction a/b with b≠0 has a finite continued fraction expansion. New proofs are also provided for the results that every field and every Euclidean domain have the idempotent matrices property.     

Invariant bilinear forms on a vertex algebra

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric. This is a generalization of the result of Li (J. Pure Appl. Algebra 96(3) (1994) 279), who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components.As an application, we introduce a notion of a radical of a vertex algebra. We prove that a radical-free vertex algebra A is non-negatively graded, and its component A₀ of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A₀-linear. We also show that in this case A is simple if and only if A₀ is a field.     

An algorithm for associative bilinear forms

We establish the equivalence between the problem of existence of associative bilinear forms and the problem of solvability in commutative power-associative finite-dimensional nil-algebras. We use the tensor product to find sufficient and necessary conditions to assure the existence of associative bilinear forms in an algebra A. The result gives us an algorithm to describe the space of associative bilinear forms for an algebra when its constants of structure γ_i,j,k for i,j,k=1,…,n are known.     

On idempotent modifications of generalized <Emphasis Type="Italic">MV</Emphasis>-algebras

The notion of idempotent modification of an algebra was introduced by Je?ek; he proved that the idempotent modification of a group is always subdirectly irreducible. In the present note we show that the idempotent modification of a generalized MV -algebra having more than two elements is directly irreducible if and only if there exists an element in A which fails to be boolean. Some further results on idempotent modifications are also proved.     

On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field.The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.     

The tetrahedron algebra and its finite-dimensional irreducible modules

Recently Terwilliger and the present author found a presentation for the three-point sl₂ loop algebra via generators and relations. To obtain this presentation we defined a Lie algebra ? by generators and relations and displayed an isomorphism from ? to the three-point sl₂ loop algebra. In this paper we classify the finite-dimensional irreducible ?-modules.