Niels Lauritzen, Jesper Funch Thomsen
We show that endomorphisms of Weyl algebras over fields of characteristic zero are flat and that birational endomorphisms are automorphisms by reducing to positive characteristic. We also give examples showing that endomorphisms of Weyl algebras are not in general flat over fields of positive characteristic.
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Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right or left form a monoidal category. The most important bimodule in this paper is the graph of an endomorphism. We prove that the graph of an endomorphism of a Weyl algebra over a field of characteristic zero is a simple bimodule. The simplicity of the tensor product of the dual graph and the graph is equivalent to the Dixmier conjecture. It is also shown how the graph construction l…
Lifts of endomorphisms of Weyl algebras modulo $p^2$
本文研究了Weyl代数在模p²下的端射影问题，证明了自同态可提升当且仅当诱导Poisson映射，并改进了Tsuchimoto的结果，指出当自同态次数小于p时，其为单射。
We prove that every endomorphism of a simple quantum generalized Weyl algebra $A$ over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of $A$. Our main result applies to minimal primitive factors of the quantized enveloping algebra of $U_q(\mathfrak{sl}_2)$ and certain minimal primitive quotients of the positive part of $U_q(\mathfrak{so}_5)$.
Hochschild cohomology and lifts of endomorphisms
研究代数自同射何时能提升为一阶平坦提升，通过关联一阶平坦提升与自同射的霍奇-希尔德上同调类，证明自同射提升当且仅当中心上的诱导自同射保持提升后的泊松结构。
We observe \cite[Proposition 4.1]{LaLe} that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra $A_1$ considered as a Poisson version of the quantum generalized Weyl algebra is constructed and its Poisson structures are studied. In particular, it is obtained a necessary and sufficient condition such that $A_1$ is Poisson simple and established that the Poisson endomorphisms of $A_1$ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.
Affine algebraic monoids as endomorphisms' monoids of finite-dimensional algebras
本文证明了任意仿射代数幺半群均可实现为某个有限维非结合代数的自同态幺半群。
We study the algebra of conformal endomorphisms $\Cend^{G,G}_n$ of a finitely generated free module $M_n$ over the coordinate Hopf algebra $H$ of a linear algebraic group $G$. It is shown that a conformal subalgebra of $\Cend_n$ acting irreducibly on $M_n$ generates an essential left ideal of $\Cend^{G,G}_n$ if enriched with operators of multiplication on elements of $H$. In particular, we describe such subalgebras for the case when $G$ is finite.
In this paper, we study the endomorphism properties of vertex operator algebras over an arbitrary field $\mathbb{F}$, with $\text{Char}(\mathbb{F}) \neq 2$. Let $V$ be a strongly finitely generated vertex operator algebra over $\mathbb{F}$, and $M$ be an irreducible admissible $V$-module. We prove that every element in $\text{End}_V(M)$ is algebraic over $\mathbb{F}$ and that $\text{End}_V(M)$ is also finite-dimensional. As an application, we prove Schur's lemma for strongly finitely generated vertex operator algebras over arbitrary algebraically closed fields, and we give a test for absolute …
Let $A_1(K)=K \langle x,y | yx-xy= 1 \rangle$ be the first Weyl algebra over a characteristic zero field $K$ and let $α$ be the exchange involution on $A_1(K)$ given by $α(x)= y$ and $α(y)= x$. The Dixmier conjecture of Dixmier (1968) asks: Is every algebra endomorphism of the Weyl algebra $A_1(K)$ an automorphism? The aim of this paper is to prove that each $α$-endomorphism of $A_1(K)$ is an automorphism. Here an $α$-endomorphism of $A_1(K)$ is an endomorphism which preserves the involution $α$. We also prove an analogue result for the Jacobian conjecture in dimension 2, called $α-JC_2$.
Let $A_1:=K\langle x, \frac{d}{dx} \rangle$ be the Weyl algebra and $\mI_1:= K\langle x, \frac{d}{dx}, \int \rangle$ be the algebra of polynomial integro-differential operators over a field $K$ of characteristic zero. The Conjecture/Problem of Dixmier (1968) [still open]: {\em is an algebra endomorphism of the Weyl algebra $A_1$ an automorphism?} The aim of the paper is to prove that {\em each algebra endomorphism of the algebra $\mI_1$ is an automorphism}. Notice that in contrast to the Weyl algebra $A_1$ the algebra $\mI_1$ is a non-simple, non-Noetherian algebra which is not a domain. Moreo…
We study endomorphisms and derivations of infinite dimensional cyclic Leibniz algebra.
We study algebra endomorphisms and derivations of some localized down-up algebras $\A$. First, we determine all the algebra endomorphisms of $\A$ under some conditions on $r$ and $s$. We show that each algebra endomorphism of $\A$ is an algebra automorphism if $r^{m}s^{n}=1$ implies $m=n=0$. When $r=s^{-1}=q$ is not a root of unity, we give a criterion for an algebra endomorphism of $\A$ to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for $\A$. We also show that each surjective algebra endomorphism of the down-up algebra $A(r+s, -rs)$ is a…
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. We investigate the structure properties of the endomorphism algebras of semi-tilting $A$-modules, and prove that the endomorphism algebras arising from the mutations of semi-tilting $A$-modules can be realized as the endomorphism algebras of BB-tilting modules.
A Hecke endomorphism algebra is a natural generalisation of the $q$-Schur algebra associated with the symmetric group to a Coxeter group. For Weyl groups, B. Parshall, L. Scott and the first author \cite{DPS,DPS4} investigated the stratification structure of these algebras in order to seek applications to representations of finite groups of Lie type. In this paper we investigate the presentation problem for Hecke endomorphism algebras associated with arbitrary Coxeter groups. Our approach is to present such algebras by quivers with relations. If $R$ is the localisation of $\mathbb Z[q]$ at the…
In this paper we prove Schur-Weyl duality between the symplectic group and Brauer algebra over an arbitrary infinite field $K$. We show that the natural homomorphism from the Brauer algebra $B_n(-2m)$ to the endomorphism algebra of tensor space $(K^{2m})^{\otimes n}$ as a module over the symplectic similitude group $GSp_{2m}(K)$ (or equivalently, as a module over the symplectic group $Sp_{2m}(K)$) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for $GSp_{2m}(K)$ to the endomorphism algebra of $(K^{2m})^{\otimes n}$ as a module over $B_n(-2m)$…
Idempotent Pre-Endomorphisms of Algebras
本文将 pre-Lie 代数中的 pre-morphism 推广至任意代数，探讨其对偶形式，并研究幂等前自同态与子伴随反交换代数半直积分解的关系。
It is proved that the Jacobian of a k-endomorphism of k[x_1,...,x_n] over a field k of characteristic zero taking every tame coordinate to a coordinate, must be a nonzero constant in k. It is also proved that the Jacobian of an R-endomorphism of A:=R[x_1,...,x_n] (where R is a polynomial ring in finite number of variables over an infinite field k), taking every R-linear coordinate of A to an R-coordinate of A, is a nonzero constant in k.
We focus in this text on the adaptation to the study of shuffles of the main combinatorial tool in the theory of free Lie algebras, namely the existence of a universal algebra of endomorphisms for tensor and other cocommutative Hopf algebras: the family of Solomon's descent algebras of type A. We show that there exists similarly a natural endomorphism algebra for commutative shuffle algebras, which is a natural extension of the Malvenuto-Reutenauer Hopf algebra of permutations, or algebra of free quasi-symmetric functions. We study this new algebra for its own, establish freeness properties, s…
We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type F Pn, which are variants of lamp-lighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra sln(k), where char(k) does not divide n affords non-trivial faithful self-similarity.
We investigate conditions under which the endomorphism ring of the Leavitt path algebra $L_{K}(E)$ possesses various ring and module-theoretical properties such as being von Neumann regular, $π$-regular, strongly $π$-regular or self-injective. We also describe conditions under which $L_{K}(E)$ is continuous as well as automorphism invariant as a right $L_{K}(E)$-module.
For an endomorphism f\inEnd{M) of a left R-module M we investigate the structure and the polynomial identities of the zero-level centralizer Cen_0(f) and the factor Cen(f)/Cen_0(f). A double zero-centralizer theorem for Cen_0(Cen_0(f)) is also formulated.
We attach to any linear endomorphism f of any vector space V a structure of prelie algebra on the shuffle algebra T(V); we describe its enveloping algebra, the dual Hopf algebra and the associated group of characters. For f=Id\_V, we find the algebra of formal diffeomorphisms, seen as a subalgebra of the Connes-Kreimer Hopf algebra of rooted trees in the context of QFT; for other well-chosen f, we obtain the groups of Fliess operators in Control Theory. An algebraic structures of these Com-Prelie Hopf algebras is carried out: gradations, group of automorphisms, subobject generated by V, etc.
A central idempotent in the endomorphism algebra of a finite lattice
本文直接构造了有限格自同态代数中的一个特定幂等元，该幂等元与所有全序子格相关联。
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$, $\mathcal {D}^b(A)$ be the bounded derived category of $A$-mod and $A^{(m)}$ be the $m$-replicated algebra of $A$. In this paper, we investigate the structure properties of endomorphism algebras arising from silting mutation in $\mathcal {D}^b(A)$ and tilting mutation in $A^{(m)}$-mod.
A commutative algebra is exact if its multiplication endomorphisms are trace-free and is Killing metrized if its Killing type trace-form is nondegenerate and invariant. A Killing metrized exact commutative algebra is necessarily neither unital nor associative. Such algebras can be viewed as commutative analogues of semisimple Lie algebras or, alternatively, as nonassociative generalizations of étale (associative) algebras. Some basic examples are described and there are introduced quantitative measures of nonassociativity, formally analogous to curvatures of connections, that serve to facilita…
We investigate polynomial endomorphisms of graph $C^*$-algebras and Leavitt path algebras. To this end, we define and analyze the coding graph corresponding to each such an endomorphism. We find an if and only if condition for the endomorphism to restrict to an automorphism of the diagonal MASA, which is stated in terms of synchronization of a certain labelling on the coding graph. We show that the dynamics induced this way on the space of infinite paths (the spectrum of the MASA) is generated by an asynchronous transducer.
Dixmier's famous question says the following: Is every algebra endomorphism of the first Weyl algebra, $A_1(F)$, where $F$ is a zero characteristic field, an automorphism? Let $α$ be the exchange involution on $A_1(F)$: $α(x)= y$, $α(y)= x$. An $α$-endomorphism of $A_1(F)$ is an endomorphism which preserves the involution $α$. Then one may ask the following question, which may be called the "$α$-Dixmier's problem $1$" or the "starred Dixmier's problem $1$": Is every $α$-endomorphism of $A_1(F)$ an automorphism?
For a matrix coalgebra $C$ over some field, we determine all small subcoalgebras of the free Hopf algebra on $C$, the free Hopf algebra with a bjective antipode on $C$, and the free Hopf algebra with antipode $S$ satisfying $S^{2d}={\rm id}$ on $C$ for some fixed $d$. We use this information to find the endomorphisms of these free Hopf algebras, and to determine the centers of the categories of Hopf algebras, Hopf algebras with bijective antipode, and Hopf algebras with antipode of order dividing 2d.
We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring $k$ with identity. These notions are applied to the study of pre-Lie $k$-algebras and, more generally, Lie-admissible $k$-algebras. Associating with any algebra $(A,\cdot)$ its sub-adjacent anticommutative algebra $(A,[-,-])$ is a functor from the category of $k$-algebras with pre-morphisms to the category of anticommutative $k$-algebras. We describe the commutator of two ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre-Lie algebras. This a…
Group graded endomorphism algebras and Morita equivalences
本文证明了特征三元组蝴蝶定理的群分次Morita等价版本，从而提供了一种从相关等价构造块扩张等价的新方法。
Construction of some non-associative algebras from Associative Algebras with an endomorphism operator, a differential operator or a left averaging operator
本文引入自同态、左平均、微分和Rota-Baxter算子等概念，在特定结合代数上构造这些线性映射，以此诱导出Lie、Pre-Lie等非结合代数结构，并给出了方阵子空间M3(R)上的实例。
A closure endomorphism of a Hilbert algebra A is a mapping that is simultaneously an endomorphism of and a closure operator on A. It is known that the set CE of all closure endomorphisms of A is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of A, anti-isomorphic to the lattice of certain closure retracts of A, and compactly generated. The set of compact elements of CE coincides with the adjoint semilattice of A, conditions under which …
Last years a number of papers were devoted to describing automorphisms of semigroups of endomorphisms of free finitely generated universal algebras of some varieties: groups, semigroups, associative commutative algebras, inverse semigroups, modules, Lie algebras and some others. All these researches were inspired by the questions prof. B. Plotkin set in connection with so called universal algebraic geometry (see for example Arxiv:math, GM/0204245). The aim of this paper is to suggest a method of describing the group AutEnd(F) for a free finitely generated algebra F of an arbitrary variety of u…
Given a maximal rigid object $T$ of the cluster tube, we determine the objects finitely presented by $T$. We then use the method of Keller and Reiten to show that the endomorphism algebra of $T$ is Gorenstein and of finite representation type, as first shown by Vatne. This algebra turns out to be the Jacobian algebra of a certain quiver with potential, when the characteristic of the base field is not 3. We study how this quiver with potential changes when $T$ is mutated. We also provide a derived equivalence classification for the endomorphism algebras of maximal rigid objects.
The aim of the paper is to extend the class of generalized Weyl algebras to a larger class of rings (they are also called {\em generalized Weyl algebras}) that are determined by two ring endomorphisms rather than one as in the case of `old' GWAs. A new class of rings, the {\em diskew polynomial rings}, is introduced that is closely related to GWAs (they are GWAs under a mild condition). The, so-called, ambiskew polynomial rings are a small subclass of the class of diskew polynomial rings. Semisimplicity criteria are given for generalized Weyl algebras and diskew polynomial rings.
A polynomial endomorphism $σ\in {\rm End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). Each Jacobian map $σ$ is extended to an endomorphism $σ$ of the Weyl algebra $A_n$. The Jacobian Conjecture (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true iff the twisted (by $σ$) $P_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. It is shown that the $A_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. Furthermore, the $A_n$-module ${}^σ P_n$ is holonomic and as a result h…
In the paper {\em The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic}, J. Pure Appl. Algebra 212 (2008), no. 10, 2320-2337, see also arXiv:math/0604477, Vladimir Bavula states the following Conjecture: (BC) Any endomorphism of a Weyl algebra (in a finite characteristic case) is a monomorphism. The purpose of this preprint is to prove BC for $A_1$, show that BC is wrong for $A_n$ when $n > 1$, and prove an analogue of $BC$ for symplectic Poisson algebras.
On endomorphism algebras of string almost gentle algebras
本文证明弦温和代数的表示型与其各类 $\mathcal{R}$-自同态代数及 Cohen-Macaulay Auslander 代数的表示型等价，揭示了这些代数类别间的深层结构联系。
We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal'cev correspondence, these results have corollaries pertaining to zeta functions enumerating normal subgroups of finite index in finitely generated nilpotent groups, most notably finitely generated free nilpotent groups of any given class.
The Hopf algebra of word-quasi-symmetric functions ($\WQSym$), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on $\WQSym$. This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret $\WQSym$ as a convolution algebra of linear endomorphisms of quasi-shuffle algebras. We then use this interpretation to study the fine structure of quas…
We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra $A$ whose global dimension $\mathop{\rm gl. dim}\nolimits A\leq 2$ and any 2-term silting complex $\mathbf{P}$ in the bounded derived category ${D^b(A)}$ of $A$, the global dimension of $\mathop{\rm End}\nolimits_{D^b(A)}(\mathbf{P})$ is at most 7. We also show that for each $n>2$, there is an algebra $A$ with $\mathop{\rm gl. dim}\nolimits A=n$ such that ${D^b(A)}$ admits a 2-term silting complex $\mathbf{P}$ with $\mathop{\rm gl. dim}\nolimits \mathop{\rm End}\n…
In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. T…
Let $G$ be a finite reductive group defined over $\mathbb{F}_q$, with $q$ a power of a prime $p$. Motivated by a problem recently posed by C. Curtis, we first develop an algorithm to express each element of $G$ into a canonical form in terms of a refinement of a Bruhat decomposition, and we then use the output of the algorithm to explicitly determine the structure constants of the endomorphism algebra of a Gelfand-Graev representation of $G$ when $G=\mathrm{PGL}_3(q)$ for an arbitrary prime $p$, and when $G=\mathrm{SO}_5(q)$ for $p$ odd.
Endomorphism algebras over commutative rings and torsion in self tensor products
本文研究交换 Noetherian 局部环上有限生成模的自同态环作用下的张量积，证明在温和条件下不可分解模的自张量积必有挠元。
We relate the endomorphism rings of certain $D$-elliptic sheaves of finite characteristic to hereditary orders in central division algebras over function fields.
Transfer of derived equivalences from subalgebras to endomorphism algebras II
本文在弱 n-角化范畴中，通过将 n-角诱导的子代数迁移到逼近序列诱导的自同态代数，建立 Φ-Auslander-Yoneda 代数子代数间的导出等价，并推广文献[BP]的构造至 n-角情形。
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