scholarly journals Jordan centralizer maps on trivial extension algebras

2020 ◽  
Vol 53 (1) ◽  
pp. 58-66
Author(s):  
Mohammad Ali Bahmani ◽  
Fateme Ghomanjani ◽  
Stanford Shateyi
Keyword(s):  
Trivial Extension ◽  
Triangular Algebra ◽  
Trivial Extension Algebra ◽  
Extension Algebra

AbstractThe structure of Jordan centralizer maps is investigated on trivial extension algebras. One may obtain some conditions under which a Jordan centralizer map on a trivial extension algebra is a centralizer map. As an application, we characterize the Jordan centralizer map on a triangular algebra.

scholarly journals Higher preprojective algebras, Koszul algebras, and superpotentials

2020 ◽  
Vol 156 (12) ◽  
pp. 2588-2627
Author(s):  
Joseph Grant ◽  
Osamu Iyama
Keyword(s):  
Trivial Extension ◽  
Koszul Algebras ◽  
Preprojective Algebra ◽  
Hereditary Algebra ◽  
Dual Pairs ◽  
Bimodule Resolution ◽  
Preprojective Algebras ◽  
Projective Resolutions ◽  
Trivial Extension Algebra ◽  
Extension Algebra

In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case, we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is $d$-hereditary, all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a $d$-hereditary algebra. This allows us to recover various known homological properties of the higher preprojective algebras and to obtain a large class of almost Koszul dual pairs of algebras. We also show that when our original algebra is Koszul there is a natural map from the quadratic dual of the higher preprojective algebra to a graded trivial extension algebra.

scholarly journals Lie Derivations on Trivial Extension Algebras

2017 ◽  
Vol 31 (1) ◽  
pp. 141-153 ◽  
Author(s):  
Amir Hosein Mokhtari ◽  
Fahimeh Moafian ◽  
Hamid Reza Ebrahimi Vishki
Keyword(s):  
Trivial Extension ◽  
Lie Derivation ◽  
Trivial Extension Algebra ◽  
Triangular Algebras ◽  
Extension Algebra ◽  
Lie Derivations

Abstract In this paper we provide some conditions under which a Lie derivation on a trivial extension algebra is proper, that is, it can be expressed as a sum of a derivation and a center valued map vanishing at commutators. We then apply our results for triangular algebras. Some illuminating examples are also included.

scholarly journals Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

2020 ◽  
Author(s):  
Piotr M. Hajac ◽  
Tomasz Maszczyk
Keyword(s):  
Cyclic Homology ◽  
Quantum Deformation ◽  
Homotopy Formula ◽  
Standard Quantum ◽  
Unital Algebra ◽  
Hopf Fibrations ◽  
Chain Homotopy ◽  
Cartan Model ◽  
Extension Algebra ◽  
Hochschild Complex

AbstractViewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

ANNIHILATOR CONDITIONS IN MATRIX AND SKEW POLYNOMIAL RINGS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250079 ◽  
Author(s):  
A. ALHEVAZ ◽  
A. MOUSSAVI
Keyword(s):  
Polynomial Rings ◽  
Trivial Extension ◽  
Skew Polynomial Ring ◽  
Matrix Rings ◽  
Skew Polynomial Rings ◽  
Armendariz Rings ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Skew Polynomial ◽  
Armendariz Ring

Let R be a ring with an endomorphism α and α-derivation δ. By [A. R. Nasr-Isfahani and A. Moussavi, Ore extensions of skew Armendariz rings, Comm. Algebra 36(2) (2008) 508–522], a ring R is called a skew Armendariz ring, if for polynomials f(x) = a0 + a1 x + ⋯ + anxn, g(x) = b0+b1x + ⋯ + bmxm in R[x; α, δ], f(x)g(x) = 0 implies a0bj = 0 for each 0 ≤ j ≤ m. In this paper, radicals of the skew polynomial ring R[x; α, δ], in terms of a skew Armendariz ring R, is determined. We prove that several properties transfer between R and R[x; α, δ], in case R is an α-compatible skew Armendariz ring. We also identify some "relatively maximal" skew Armendariz subrings of matrix rings, and obtain a necessary and sufficient condition for a trivial extension to be skew Armendariz. Consequently, new families of non-reduced skew Armendariz rings are presented and several known results related to Armendariz rings and skew polynomial rings will be extended and unified.

scholarly journals Generalized Higher Derivations on Lie Ideals of Triangular Algebras

2017 ◽  
Vol 25 (1) ◽  
pp. 35-53
Author(s):  
Mohammad Ashraf ◽  
Nazia Parveen ◽  
Bilal Ahmad Wani
Keyword(s):  
Commutative Ring ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Triangular Algebra ◽  
Higher Derivation ◽  
Square Closed Lie Ideal ◽  
Triangular Algebras

Abstract Let be the triangular algebra consisting of unital algebras A and B over a commutative ring R with identity 1 and M be a unital (A; B)-bimodule. An additive subgroup L of A is said to be a Lie ideal of A if [L;A] ⊆ L. A non-central square closed Lie ideal L of A is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on A, every generalized Jordan triple higher derivation of L into A is a generalized higher derivation of L into A.

The fundamental groups of a triangular algebra

1996 ◽  
Vol 24 (1) ◽  
pp. 187-208 ◽  
Author(s):  
Ibrahim Assem ◽  
José Antonio de la Peña
Keyword(s):  
Fundamental Groups ◽  
Triangular Algebra

scholarly journals EXTENSION ALGEBRAS OF CUNTZ ALGEBRA, II

2009 ◽  
Vol 80 (1) ◽  
pp. 83-90 ◽  
Author(s):  
SHUDONG LIU ◽  
XIAOCHUN FANG
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Cuntz Algebra ◽  
Infinite Dimensional ◽  
Algebra Ii ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
K Theory ◽  
Extension Algebra

AbstractIn this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

scholarly journals Zero-Sum Coefficient Derivations in Three Variables of Triangular Algebras

2016 ◽  
Vol 8 (5) ◽  
pp. 37
Author(s):  
Youngsoo Kim ◽  
Byunghoon Lee
Keyword(s):  
Standard Form ◽  
Sufficient Conditions ◽  
Multilinear Polynomial ◽  
Triangular Algebra ◽  
Triangular Algebras ◽  
Zero Sum ◽  
Lie Derivations

Under mild assumptions Benkovi\v{c} showed that an $f$-derivation of a triangular algebra is a derivation when the sum of the coefficients of the multilinear polynomial $f$ is nonzero. We investigate the structure of $f$-derivations of triangular algebras when $f$ is of degree 3 and the coefficient sum is zero. The zero-sum coeffient derivations include Lie derivations (degree 2) and Lie triple derivations (degree 3), which have been previously shown to be not necessarily derivations but in standard form, i.e., the sum of a derivation and a central map. In this paper, we present sufficient conditions on the coefficients of $f$ to ensure that any $f$-derivations are derivations or are in standard form.<br /><br />

scholarly journals The fundamental group of an algebra with a strongly simply connected Galois covering

2019 ◽  
Vol 18 (03) ◽  
pp. 1950052
Author(s):  
Diane Castonguay ◽  
Claudia Chaio ◽  
Sonia Trepode
Keyword(s):  
Fundamental Group ◽  
Galois Covering ◽  
Triangular Algebra ◽  
Simply Connected

In this work, we prove that if a triangular algebra [Formula: see text] admits a strongly simply connected universal Galois covering for a given presentation, then the fundamental group associated to this presentation is free.

scholarly journals Representation-finite trivial extension algebras

1984 ◽  
Vol 33 (3) ◽  
pp. 235-242 ◽  
Author(s):  
Ibrahim Assem ◽  
Dieter Happel ◽  
Oscar Roldán
Keyword(s):  
Trivial Extension
Sign in / Sign up

Export Citation Format

Share Document