Jorge A. Guccione, Juan J. Guccione
Let k a characteristic zero field. We give a characterization for the finite quiver k-algebras, based on double derivations. More precisely, we prove that if an associative and unitary k-algebra have a family of double derivations satisfying suitable conditions, then it is (canonically isomorphic to) a quiver algebra. This is the non-commutative version of a result of D. Wright.
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We consider algebras defined from quivers with relations that are k-th order derivations of a superpotential, generalizing results of Dubois-Violette to the quiver case. We give a construction compatible with Morita equivalence, and show that many important algebras arise in this way, including McKay correspondence algebras for GL_n for all n, and four-dimensional Sklyanin algebras. More generally, we show that any N-Koszul, (twisted) Calabi-Yau algebra must have a (twisted) superpotential, and construct its minimal resolution in terms of derivations of the (twisted) superpotential. This yield…
Let $K$ be a field and $Γ$ a finite quiver without oriented cycles. Let $Λ$ be the path algebra $K(Γ, ρ)$ and let $\mathscr{D}(Λ)$ be the dual extension of $Λ$. In this paper, we prove that each Lie derivation of $\mathscr{D}(Λ)$ is of the standard form.
Without the faithful assumption, we prove that every Jordan derivation on a class of path algebras of quivers without oriented cycles is a derivation and that every Lie derivation on such kinds of algebras is of the standard form.
We present the Gerstenhaber algebra structure on the Hochschild cohomology of Koszul algebras defined by quivers and relations using the idea of homotopy liftings. E.L. Green, G. Hartman, E.N. Marcos and O. Solberg provided a canonical way of constructing a minimal projective bimodule resolution of a Koszul quiver algebra. The resolution has a comultiplicative structure which we use to define homotopy lifting maps. We first present a short example that demonstrates the theory. We then study the Gerstenhaber algebra structure on Hochschild cohomology of a family of bound quiver algebras, some m…
We consider the action of a semisimple Hopf algebra $H$ on an $m$-Koszul Artin-Schelter regular algebra $A$. Such an algebra $A$ is a derivation-quotient algebra for some twisted superpotential $\mathsf{w}$, and we show that the homological determinant of the action of $H$ on $A$ can be easily calculated using $\mathsf{w}$. Using this, we show that the smash product $A\,\#\,H$ is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra $Λ$ to which $A\,\#\,H$ is Morita equivalent, generalising a result of Bocklandt-Schedler-Wemyss. We also show how $Λ$ can be u…
We construct Artin algebras with vanishing first Hochschild Cohomology but a loop in their ordinary quiver. Some authors guessed earlier that such examples could not exist.
In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of double Poisson-Lichnerowicz cohomology for double Poisson algebras, and give some elementary properties. We introduce the notion of a linear double Poisson tensor on a quiver and show that it induces the structure of a finite dimensional algebra on the vector spaces V_v generated by the loops in the vertex v. We show that the Hochschild cohomology of the associ…
Quiver down-up algebras of type A
本文推广了Benkart和Roby定义的下上代数，证明其为Noetherian分段域，并探讨了分级有向图下上代数的同构问题。
We show that the Hochschild cohomology of a monomial algebra over a field of characteristic zero vanishes from degree two if the first Hochschild cohomology is semisimple as a Lie algebra. We also prove that first Hochschild cohomology of a radical square zero algebra is reductive as a Lie algebra. In the case of the multiple loops quiver, we obtain the Lie algebra of square matrices of size equal to the number of loops.
We study classical R-matrices D for Lie algebras L such that D is also a derivation of L. This yields derivation double Lie algebras (L,D). The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.
We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form $A = \kk Q/I$, where $Q$ is a quiver and $I$ is an ideal of relations coming from taking partial derivatives of a twisted superpotential on $Q$. We define the type $(M, P, d)$ of such an algebra $A$, where $M$ is the incidence matrix of the quiver, $P$ is the permutation matrix giving the action of the Nakayama automorphism of $A$ on the vertices of the quiver, and $d$ is the degree of the superpotential. We study the question of what possible types can occur under the additional as…
This article studies the Lie algebra $Diff(KΓ)$ of derivations on the path algebra $KΓ$ of a quiver $Γ$ and the Lie algebra on the first Hochschild cohomology group $H^1(KΓ)$. We relate these Lie algebras to the algebraic and combinatorial properties of the path algebra. Characterizations of derivations on a path algebra are obtained, leading to a canonical basis of $Diff(KΓ)$ and its Lie algebra properties. Special derivations are associated to the vertices, arrows and faces of a quiver, and the concepts of a connected matrix and boundary matrix are introduced to study the relations among the…
Double derivations of $n-$Hom-Lie color algebras
本文研究 $n-$Hom-Lie color 代数的双导子代数，刻画其与普通导子代数的关系，并证明内导子理想性、中心化子平凡性及无中心完备情形下三导子等于导子。
We classify Hopf actions of Taft algebras T(n) on path algebras of quivers, in the setting where the quiver is loopless, finite, and Schurian. As a corollary, we see that every quiver admitting a faithful Z_n-action (by directed graph automorphisms) also admits inner faithful actions of a Taft algebra. Several examples for actions of the Sweedler algebra T(2) and for actions of T(3) are presented in detail. We then extend the results on Taft algebra actions on path algebras to actions of the Frobenius-Lusztig kernel u_q(sl_2), and to actions of the Drinfeld double of T(n).
Nakayama automorphisms of graded double Ore extensions of Koszul Artin-Schelter regular algebras with nontrivial skew derivations
本文研究Koszul Artin-Schelter正则代数的分次双重Ore扩张，通过构造极小自由预解证明其保持Koszul性质，并引入$ς$-散度同调不变量精确刻画中山自同构，进而构造扭曲超势实现导子商代数同构。
We construct an explicit projective bimodule resolution for the Leavitt path algebra of a row-finite quiver. We prove that the Leavitt path algebra of a row-countable quiver has Hochschild cohomolgical dimension at most one, that is, it is quasi-free in the sense of Cuntz-Quillen. The construction of the resolution relies on an explicit derivation of the Leavitt path algebra.
To a tree of semi-simple algebras we associate a qurve (or formally smooth algebra) S. We introduce a Zariski- and etale quiver describing the finite dimensional representations of S. In particular, we show that all quotient varieties of the etale quiver have a natural Poisson structure induced by a double Poisson algebra structure on a certain universal localization of its path algebra. Explicit calculations are included for the group algebras of the arithmetic groups (P)SL2(Z) and GL2(Z) but the methods apply as well to congruence subgroups.
Double (quasi-)Poisson algebras were introduced by Van den Bergh as non-commutative analogues of algebras endowed with a (quasi-)Poisson bracket. In this work, we provide a study of morphisms of double (quasi-)Poisson algebras, which we relate to the $H_0$-Poisson structures of Crawley-Boevey. We prove in particular that the double (quasi-)Poisson algebra structure defined by Van den Bergh for an arbitrary quiver only depends upon the quiver seen as an undirected graph, up to isomorphism. We derive from our results a representation theoretic description of action-angle duality for several clas…
We described $δ$-derivations and $δ$-superderivations of simple Jordan superalgebra <<KKM Double>> (also known as superalgebra of Jordan brackets) and unital simple finite-dimensional Jordan superalgebras over algebraic closed fields with characteristic $p\neq2$. As a consequence, we received relationship between non-trivial $δ$-derivations and specialty of simple superalgebra <<KKM Double>>. We constructed new examples of non-trivial 1/2-derivations of Jordan superalgebras of vector type.
The paper of Domokos is "Invariants of quivers and wreath products," and as in that paper we use Regev's double centralizer theorem for wreath products to study trace identities and invariant theory. In addition to extending to other algebras and groups, we also compute Poincare series and prove embedding theorems.
In \cite{JS} Jensen and Su constructed 0-Schur algebras on double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver approach naturally gives rise to a new class of algebras. That is, the path algebras defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters $\ut$. In particular, when all the entries of $\ut$ are 1, we have 0-Schur algerbas. When all the entries of $\ut$ are zero, we obtain a class of degenerate 0-Schur algebras. We prove that…
We study the subregular $J$-ring $J_C$ of a Coxeter system $(W,S)$, a subring of Lusztig's $J$-ring. We prove that $J_C$ is isomorphic to a quotient of the path algebra of the double quiver of $(W,S)$ by a suitable ideal that we associate to a family of Chebyshev polynomials. As applications, we use quiver representations to study the category mod-$A_K$ of finite dimensional right modules of the algebra $A_K=K\otimes_\Z J_C$ over an algebraically closed field $K$ of characteristic zero. Our results include classifications of Coxeter systems for which mod-$A_K$ is semisimple, has finitely many …
We give a description of the bimodule of double derivations DDer(S) of a finite dimensional semi-simple algebra S and its double Schouten bracket in terms of a quiver. This description is used to determine which degree two monomials in T_SDDer(S) induce double Poisson brackets on S. In case S = C^n, a criterion for any degree two element to give a double Poisson bracket is deduced. For S = C^n and S' = C^m, the induced Poisson bracket on the variety of isomorphism classes of semi-simple representations iss_n(S*T) of the free product S*T is given.
Double shuffle Lie algebra and special derivations
本文研究了双重洗牌李代数$\mathfrak{dmr}_0$的性质，证明其包含于特殊导出子代数，并探讨了相关双重洗牌方案的结构。
We investigate the structure of the double Ringel-Hall algebras associated with cyclic quivers and its connections with quantum loop algebras of $\mathfrak{gl}_n$, affine quantum Schur algebras and affine Hecke algebras. This includes their Drinfeld-Jimbo type presentation, affine quantum Schur-Weyl reciprocity, representations of affine quantum Schur algebras, and connections with various existing works by Lusztig, Varagnolo-Vasserot, Schiffmann, Hubery, Chari-Pressley, Frenkel-Mukhin, etc. We will also discuss conjectures on a realization of Beilinson-Lusztig-MacPherson type and Lusztig type…
We give a new construction of cyclic homology of an associative algebra A that does not involve Connes' differential. Our approach is based on an extended version of the complex ΩA, of noncommutative differential forms on A, and is similar in spirit to the de Rham approach to equivariant cohomology. Indeed, our extended complex maps naturally to the equivariant de Rham complex of any representation scheme Rep A. We define cyclic homology as the cohomology of the total complex (ΩA)[t], d+t \cdot i, arising from two anti-commuting differentials, d and i, on ΩA. The differential d, that replaces …
Functorial constructions related to double Poisson vertex algebras
本文通过jet代数构造为任意双重泊松代数生成双重泊松顶点代数，证明其与表示函子相容，并比较泊松约化与哈密顿约化范畴，给出双重箭图的实例。
We defined generalized δ-derivations of algebra A as linear mapping χassociated with usual δ-derivation ϕby the rule χ(xy)=δ(χ(x)y+xϕ(y))=δ(ϕ(x)y+xχ(y)) for any x,y \in A. We described generalized δ-derivations of prime alternative algebras, prime Lie algebras and superalgebras, unital algebras, and semisimple finite-dimensional Jordan superalgebras. In this cases we proved that generalized δ-derivation is a generalized derivation or δ-derivation. After that we described δ-superderivations of superalgebras <<KKM Double>>, arising from prime alternative algebras, prime Lie algebras and superalg…
The goal of this paper is to prove that if Q is a connected non-Dynkin quiver then the preprojective algebra of Q over any field k is Koszul, and has Hilbert series 1/(1-Ct+t^2), where C is the adjacency matrix of the double of Q. (This result, in somewhat less general formulations, was previously obtained by Martinez-Villa and Malkin-Ostrik-Vybornov). We also prove a similar result for the partial preprojective algebra of any connected quiver Q, associated to a subset J of the set I of vertices of Q (by definition, this is the quotient of the path algebra of the double by the preprojective al…
Quasi-Poisson varieties from double quasi-Poisson algebras in types $B,C,D$
本文通过引入对合反自同构，将双拟泊松括号推广到任意半单基上的扭表示空间，并应用于箭图以理解扭（局部乘性）箭图簇上的泊松结构，同时恢复了正交/辛群特征簇的泊松结构并引入修正的Kontsevich系统。
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