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In this paper, we introduce and study differential graded (DG for short) polynomial algebras. In brief, a DG polynomial algebra $mathcal{A}$ is a connected cochain DG algebra such that its underlying graded algebra $mathcal{A}^{#}$ is a polynomial algebra $mathbb{k}[x_1,x_2,cdots, x_n]$ with $|x_i|=1$, for any $iin {1,2,cdots, n}$. We describe all possible differential structures on DG polynomial algebras; compute their DG automorphism groups; study their isomorphism problems; and show that they are all homologically smooth and Gorestein DG algebras. Furthermore, it is proved that the DG polynomial algebra $mathcal{A}$ is a Calabi-Yau DG algebra when its differential $partial_{mathcal{A}} eq 0$ and the trivial DG polynomial algebra $(mathcal{A}, 0)$ is Calabi-Yau if and only if $n$ is an odd integer.
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Let G be any group and F an algebraically closed field of characteristic zero. We show that any G-graded finite dimensional associative G-simple algebra over F is determined up to a G-graded isomorphism by its G-graded polynomial identities. This result was proved by Koshlukov and Zaicev in case G is abelian.
In this paper, we compute all possible differential structures of a $3$-dimensional DG Sklyanin algebra $mathcal{A}$, which is a connected cochain DG algebra whose underlying graded algebra $mathcal{A}^{#}$ is a $3$-dimensional Sklyanin algebra $S_{a,b,c}$. We show that there are three major cases depending on the parameters $a,b,c$ of the underlying Sklyanin algebra $S_{a,b,c}$: (1) either $a^2 eq b^2$ or $c eq 0$, then $partial_{mathcal{A}}=0$; (2) $a=-b$ and $c=0$, then the $3$-dimensional DG Sklyanin algebra is actually a DG polynomial algebra; and (3) $a=b$ and $c=0$, then the DG Sklyanin algebra is uniquely determined by a $3times 3$ matrix $M$. It is worthy to point out that case (2) has been systematically studied in cite{MGYC} and case (3) is just the DG algebra $mathcal{A}_{mathcal{O}_{-1}(k^3)}(M)$ in cite{MWZ}. We solve the problem on how to judge whether a given $3$-dimensional DG Sklyanin algebra is Calabi-Yau.
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different classes are constructed. Applications to quantum mechanical systems including supersymmetric quantum mechanics are discussed
We construct four families of Artin-Schelter regular algebras of global dimension four. Under some generic conditions, this is a complete list of Artin-Schelter regular algebras of global dimension four that are generated by two elements of degree 1. These algebras are also strongly noetherian, Auslander regular and Cohen-Macaulay. One of the main tools is Kellers higher-multiplication theorem on A-infinity Ext-algebras.
This paper concerns homological notions of regularity for noncommutative algebras. Properties of an algebra $A$ are reflected in the regularities of certain (complexes of) $A$-modules. We study the classical Tor-regularity and Castelnuovo-Mumford regularity, which were generalized from the commutative setting to the noncommutative setting by J{o}rgensen and Dong-Wu. We also introduce two new numerical homological invariants: concavity and Artin-Schelter regularity. Artin-Schelter regular algebras occupy a central position in noncommutative algebra and noncommutative algebraic geometry, and we use these invariants to establish criteria which can be used to determine whether a noetherian connected graded algebra is Artin-Schelter regular.
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