We construct double cross biproduct and bi-cycle bicrossproduct Lie bialgebras from braided Lie bialgebras. The main result generalizes Majids matched pairs of Lie algebras, Drinfelds quantum double, and Masuokas cross product Lie bialgebras.
Consider any representation $phi$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $hat{S}(g^*)$ of its dual. Consider the tensor product of $hat{S}(g^*)$ and the exterior algebra $Lambda(g)$. We show that the
representation $phi$ extends canonically to the representation $tildephi$ of that tensor product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncommutative differentials and coordinates are formal power series in partial derivatives. The square of the corresponding exterior derivative is zero like in the classical case, but the Leibniz rule is deformed.
We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $ell(B(I,G))$ and $ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, w
e show that if $G$ is an amenable group, then for a wide class of Banach $ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.
Let $n geq 2$ be an integer. An emph{$n$-potent} is an element $e$ of a ring $R$ such that $e^n = e$. In this paper, we study $n$-potents in matrices over $R$ and use them to construct an abelian group $K_0^n(R)$. If $A$ is a complex algebra, there i
s a group isomorphism $K_0^n(A) cong bigl(K_0(A)bigr)^{n-1}$ for all $n geq 2$. However, for algebras over cyclotomic fields, this is not true in general. We consider $K_0^n$ as a covariant functor, and show that it is also functorial for a generalization of homomorphism called an emph{$n$-homomorphism}.
We provide a full characterization of the oblique projector $U(VU)^+V$ in the general case where the range of $U$ and the null space of $V$ are not complementary subspaces. We discuss the new result in the context of constrained least squares minimization.
The relationship between Heyting algebras (HA) and semirings is explored. A new class of HAs called Symmetric Heyting algebras (SHAs) is proposed, and a necessary condition on SHAs to be consider semirings is given. We define a new mathematical famil
y called Heyting structures, which are similar to semirings, but with Heyting-algebra operators in place of the usual arithmetic operators usually seen in semirings. The impact of the zero-sum free property of semirings on Heyting structures is shown as also the condition under which it is possible to extend one Heyting structure to another. It is also shown that the union of two or more sets forming Heyting structures is again a Heyting structure, if the operators on the new structure are suitably derived from those of the component structures. The analysis also provides a sufficient condition such that the larger Heyting structure satisfying a monotony law implies that the ones forming the union do so as well.
If A is a cocommutative algebra with coproduct, then so is the smash product algebra of a symmetric algebra Sym(V) with A, where V is an A-module. Such smash product algebras, with A a group ring or a Lie algebra, have families of deformations that h
ave been studied widely in the literature; examples include symplectic reflection algebras and infinitesimal Hecke algebras. We introduce a family of deformations of these smash product algebras for general A, and characterize the PBW property. We then characterize the Jacobi identity for grouplike algebras (that include group rings and the nilCoxeter algebra), and precisely identify the PBW deformations in the example where A is the nilCoxeter algebra. We end with the more prominent case - where A is a Hopf algebra. We show the equivalence of sever
Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = cap lbrace I colon I text{is an ideal of} R text{and} x in IM rbrace $. $M$ is said to be a content $R$-module if $x in c(x)M $, for all $x in M$. $B$ is c
alled a content $R$-algebra, if it is a faithfully flat and content $R$-module and it satisfies the Dedekind-Mertens content formula. In this article, we prove some new results for content modules and algebras by using ideal theoretic methods.
The results of Iv. Prodanov on abstract spectra and separative algebras were announced in the journal Trudy Mat. Inst. Steklova, 154, 1983, 200--208, but their proofs were never written by him in the form of a manuscript, preprint or paper. Since the
untimely death of Ivan Prodanov withheld him from preparing the full version of this paper and since, in our opinion, it contains interesting and important results, we undertook the task of writing a full version of it and thus making the results from it known to the mathematical community. So, the aim of this paper is to supply with proofs the results of Ivan Prodanov announced in the cited above paper, but we added also a small amount of new results. The full responsibility for the correctness of the proofs of the assertions presented below in this work is taken by us; just for this reason our names appear as authors of the present paper.
In this paper we describe some Leibniz algebras whose corresponding Lie algebra is four-dimensional Diamond Lie algebra $mathfrak{D}$ and the ideal generated by the squares of elements (further denoted by $I$) is a right $mathfrak{D}$-module. Using d
escription cite{Cas} of representations of algebra $mathfrak{D}$ in $mathfrak{sl}(3,{mathbb{C}})$ and $mathfrak{sp}(4,{mathbb{F}})$ where ${mathbb{F}}={mathbb{R}}$ or ${mathbb{C}}$ we obtain the classification of above mentioned Leibniz algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended to the case of the algebra $mathfrak{D}.$ Classification of Leibniz algebras with corresponding Lie algebra $mathfrak{D}$ and with the ideal $I$ as a Fock right $mathfrak{D}$-module is presented. The linear integrable deformations in terms of the second cohomology groups of obtained finite-dimensional Leibniz algebras are described. Two computer programs in Mathematica 10 which help to calculate for a given Leibniz algebra the general form of elements of spaces $BL^2$ and $ZL^2$ are constructed, as well.
