On Eilenberg–Moore algebras induced by chains

Let S be a fixed topological space. The contravariant Hom functor given by C(X) = has an adjoint specified, on sets, by P(A)= S ^A and the composite, , is a Monad on the category of sets. In this paper we characterize the category of Eilenberg–Moore Algebras associated with M in the special case where S is a linearly ordered space in its specialization order. The characterization is presented in terms of the notion of a dual frame which admits a C(S)-action. Received June 11, 1998, accepted in final form March 2, 1999     

Calabi–Yau Algebras Viewed as Deformations of Poisson Algebras

From any algebra A defined by a single non-degenerate homogeneous quadratic relation f, we prove that the quadratic algebra B defined by the potential w?=?fz is 3-Calabi–Yau. The algebra B can be viewed as a 3-Calabi–Yau completion of Keller. The algebras A and B are both Koszul. The classification of the algebras B in three generators, i.e., when A has two generators, leads to three types of algebras. The second type (the most interesting one) is viewed as a deformation of a Poisson algebra S whose Poisson bracket is non-diagonalizable quadratic. Although the potential of S has non-isolated singularities, the homology of S is computed. Next the Hochschild homology of B is obtained.     

Torsion in the cohomology of finite loop spaces and the Eilenberg–Moore spectral sequence

We use the Eilenberg–Moore spectral sequence to study torsion phenomena in the integral cohomology of finite loop spaces with maximal torus generalizing some results for compact Lie groups due to Kac.     

Ringel–Hall Algebras of Quotient Algebras of Path Algebras

We determine the generating relations for Ringel–Hall algebras associated with quotient algebras of path algebras of Dynkin and tame quivers, and investigate their connection with composition subalgebras.     

Cotorsion Pairs and Cartan–Eilenberg Categories

Cartan–Eilenberg categories were recently introduced by Guillén Santos, Navarro Aznar, Pascual and Roig. In the present paper, we give a method of constructing Cartan–Eilenberg categories for abelian categories, based on cotorsion pairs. In particular, we recover the left Cartan–Eilenberg structure for (un)bounded below chain complexes of modules (Pascual, Collect Math 63(2):203–216, 2012) and extend it to more general categories, including the category of (quasi-coherent)sheaves over a projective scheme.     

Algebras of Quotients of Jordan–Lie Algebras

In this article, we introduce the notion of algebra of quotients of a Jordan–Lie algebra. Properties such as semiprimeness or primeness can be lifted from a Jordan–Lie algebra to its algebras of quotients. Finally, we construct a maximal algebra of quotients for every semiprime Jordan–Lie algebra.     

Simple Poisson–Farkas Algebras and Ternary Filippov Algebras

We establish connection between the differentiably simple associative commutative algebras with unity and the simple Filippov algebras.     

Complex Kumjian–Pask Algebras

LetΛbe a row-fnitek-graph without sources.We investigate the relationship between the complex Kumjian-Pask algebra KPC(Λ)and graph algebra C(Λ).We identify situations in which the Kumjian-Pask algebra is equal to the graph algebra,and the conditions in which the Kumjian-Pask algebra is fnite-dimensional.     

Monoidal Hom–Hopf Algebras

Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras, and Lie algebras.     

Polynomial-Time Homology for Simplicial Eilenberg–MacLane Spaces

In an earlier paper of ?adek, Vok?ínek, Wagner, and the present authors, we investigated an algorithmic problem in computational algebraic topology, namely, the computation of all possible homotopy classes of maps between two topological spaces, under suitable restriction on the spaces. We aim at showing that, if the dimensions of the considered spaces are bounded by a constant, then the computations can be done in polynomial time. In this paper we make a significant technical step towards this goal: we show that the Eilenberg–MacLane space $K(\mathbb{Z},1)$ , represented as a simplicial group, can be equipped with polynomial-time homology (this is a polynomial-time version of effective homology considered in previous works of the third author and co-workers). To this end, we construct a suitable discrete vector field, in the sense of Forman’s discrete Morse theory, on $K(\mathbb{Z},1)$ . The construction is purely combinatorial and it can be understood as a certain procedure for reducing finite sequences of integers, without any reference to topology. The Eilenberg–MacLane spaces are the basic building blocks in a Postnikov system, which is a “layered” representation of a topological space suitable for homotopy-theoretic computations. Employing the result of this paper together with other results on polynomial-time homology, in another paper we obtain, for every fixed k, a polynomial-time algorithm for computing the kth homotopy group π _k (X) of a given simply connected space X, as well as the first k stages of a Postnikov system for X, and also a polynomial-time version of the algorithm of ?adek et al. mentioned above.     

Cohen–Macaulay Multi–Rees Algebras

Let A be a local ring, and let I ₁,...,I _r A be ideals of positive height. In this article we compare the Cohen–Macaulay property of the multi–Rees algebra R _A (I ₁,...,I _r ) to that of the usual Rees algebra R _A (I ₁ ··· I _r ) of the product I ₁ ··· I _r . In particular, when the analytic spread of I ₁ ··· I _r is small, this leads to necessary and sufficient conditions for the Cohen–Macaulayness of R _A (I ₁,...,I _r ). We apply our results to the theory of joint reductions and mixed multiplicities.     

Rota–Baxter Hom-Lie–Admissible Algebras

We study Hom-type analogs of Rota–Baxter and dendriform algebras, called Rota–Baxter G-Hom–associative algebras and Hom-dendriform algebras. Several construction results are proved. Free algebras for these objects are explicitly constructed. Various functors between these categories, as well as an adjunction between the categories of Rota–Baxter Hom-associative algebras and of Hom-(tri)dendriform algebras, are constructed.     

Igusa–Todorov Endomorphism Algebras

We study polynomial identities of algebras with adjoined external unit. For a wide class of algebras we prove that adjoining external unit element leads to increasing of PI-exponent precisely to 1. We also show that any real number from the interval [2,3] can be realized as PI-exponent of some unital algebra.     

Cartan–Eilenberg complexes with respect to cotorsion pairs

Let ${(\mathcal{A}, \mathcal{B})}$ be a cotorsion pair in R-Mod. We introduce and study the notion of Cartan–Eilenberg- ${\mathcal{A}}$ complexes. Specifically, we establish some relationships between Cartan–Eilenberg- ${\mathcal{A}}$ complexes and DG- ${\mathcal{A}}$ complexes. Using these relationships, some applications are given.     

Algebra Structures on the Twisted Eilenberg–Zilber Theorem

Let G, G′, and G ×_τ G′ be three simplicial groups (not necessarily abelian) and C _N (G) ? _t  C _N (G′) be the “twisted” tensor product associated to C _N (G ×_τ G′) by the twisted Eilenberg–Zilber theorem. Here we prove that the pair (C _N (G) ? _t  C _N (G′), μ) is a DGA-algebra where μ is the standard product of C _N (G) ? C _N (G′). Furthermore, the injection of the twisted Eilenberg–Zilber contraction is a DGA-algebra morphism and the projection and the homotopy operator satisfy other weaker multiplicative properties.     

On Strong Kadison–Singer Algebras

We show that many Kadison–Singer algebras are maximal triangular in all algebras containing them although their definition requires the maximality taken in the class of reflexive algebras. Diagonal-trivial maximal non self-adjoint subalgebras of matrix algebras with lower dimensions are classified.