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We provide a construction of minimal injective resolutions of simple comodules of path coalgebras of quivers with relations. Dual to Calabi-Yau condition of algebras, we introduce the Calabi-Yau condition to coalgebras. Then we give some descriptions of Calabi-Yau coalgebras with lower global dimensions. An appendix is included for listing some properties of cohom functors.
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The Calabi-Yau property of cocommutative Hopf algebras is discussed by using the homological integral, a recently introduced tool for studying infinite dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group algebra of a universal enveloping algebra of a finite dimensional Lie algebra $g$ with a finite subgroup $G$ of automorphisms of $g$ is Calabi-Yau if and only if the universal enveloping algebra itself is Calabi-Yau and $G$ is a subgroup of the special linear group $SL(g)$. The Noetherian cocommutative Calabi-Yau Hopf algebras of dimension not larger than 3 are described. The Calabi-Yau property of Sridharan enveloping algebras of finite dimensional Lie algebras is also discussed. We obtain some equivalent conditions for a Sridharan enveloping algebra to be Calabi-Yau, and then partly answer a question proposed by Berger. We list all the nonisomorphic 3-dimensional Calabi-Yau Sridharan enveloping algebras.
We prove that multiplicative preprojective algebras, defined by Crawley-Boevey and Shaw, are 2-Calabi-Yau algebras, in the case of quivers containing unoriented cycles. If the quiver is not itself a cycle, we show that the center is trivial, and hence the Calabi-Yau structure is unique. If the quiver is a cycle, we show that the algebra is a non-commutative crepant resolution of its center, the ring of functions on the corresponding multiplicative quiver variety with a type A surface singularity. We also prove that the
We give an explicit formula showing how the double Poisson algebra introduced in cite{VdB} appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on $Aoplus A^*$. Specific part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structures. The result holds for any associative algebra $A$ and emphasizes the special role of the fourth component of a pre-Calabi-Yau structure in this respect. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson brackets on representation spaces $({rm Rep}_n A)^{Gl_n}$ for any associative algebra $A$.
Let $H$ and $L$ be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if $H$ is a twisted Calabi-Yau (CY) Hopf algebra, then $L$ is a twisted CY algebra when it is homologically smooth. Especially, if $H$ is a Noetherian twisted CY Hopf algebra and $L$ has finite global dimension, then $L$ is a twisted CY algebra.
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.
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