Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS（斯特拉斯堡大学）；Université Sorbonne Paris Nord, Laboratoire de Géométrie, Analyse et Applications, LAGA, CNRS, UMR 7539（索邦巴黎北大学）
Vladimir Dotsenko, Oisín Flynn-Connolly
备注：
15 pages, substantially revised version including a general PBW type criterion for modules of Kähler differentials
发表信息：
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 176 , Issue 2 , March 2024 , pp. 441 - 458
我们给出了三种出现在预李代数理论中的Schur函子的显式组合描述。其中第一个导致了给定李代数的universal enveloping pre-Lie代数的底层向量空间的函子性描述，强化了Segal的PBW定理。另外两个Schur函子则提供了给定预李代数的universal multiplicative enveloping algebra和Kähler微分模的底层向量空间的函子性描述。此类描述的一个重要结果是，预李代数的cohomology with coefficients in a module被解释为对于universal multiplicative enveloping algebra的模块范畴的导出函子。
英文摘要：
We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the PBW theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying vector spaces of the universal multiplicative enveloping algebra and of the module of Kähler differentials of a given pre-Lie algebra. An important consequence of such descriptions is an interpretation of the cohomology of a pre-Lie algebra with coefficients in a module as a derived functor for the category of modules over the universal multiplicative enveloping algebra.
We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz-Happel's theorem, we can describe singularity categories of certain non-Gorenstein rings via the stable category of maximal Cohen-Macaulay modules. Three concrete examples of finite-dimensional algebras with the same singularity category are discussed.
The theory of Schur functors provides a powerful and elegant approach to the representation theory of GL_n - at least to the so-called polynomial representations - especially to questions about how the theory varies with n. We develop parallel theories that apply to other classical groups and to non-polynomial representations of GL_n. These theories can also be viewed as linear analogs of the theory of FI-modules.
We introduce categories of homogeneous strict polynomial functors, $\Pol^\I_{d,\k}$ and $\Pol^\II_{d,\k}$, defined on vector superspaces over a field $\k$ of characteristic not equal 2. These categories are related to polynomial representations of the supergroups $GL(m|n)$ and Q(n), respectively. In particular, we prove an equivalence between $\Pol^\I_{d,\k}$, $\Pol^\II_{d,\k}$ and the category of finite dimensional supermodules over the Schur superalgebra $\Sc(m|n,d)$, $\Qc(n,d)$ respectively provided $m,n \ge d$. We also discuss some aspects of Sergeev duality from the viewpoint of the categ…
Let $R$ be an Artin algebra and $e$ an idempotent of $R$. Assume that ${\rm Tor}_i^{eRe}(Re,G)=0$ for any $G\in{\rm GProj} eRe$ and $i$ sufficiently large. Necessary and sufficient conditions are given for the Schur functor $S_e$ to induce a triangle-equivalence $\mathbb{D}_{def}(R)\simeq\mathbb{D}_{def}(eRe)$. Combine this with a result of Psaroudakis-Skartsaterhagen-Solberg [29], we provide necessary and sufficient conditions for the singular equivalence $\mathbb{D}_{sg}(R)\simeq\mathbb{D}_{sg}(eRe)$ to restrict to a triangle-equivalence $\underline{{\rm GProj} R}\simeq\underline{{\rm GProj}…
It is important for applications of Homological Algebra in Representation Theory to have control over the behaviour of (minimal) projective resolutions under various functors. In this article we describe three broad families of functors that preserve such resolutions. We will use these results in our work on Representation Theory of Schur algebras.
For a free presentation $0 \to R \to F \to G \to 0$ of a Leibniz algebra $G$, the Baer invariant ${\cal M}^{\sf Lie}(G) = \frac{R \cap [F, F]_{Lie}}{[F, R]_{Lie}}$ is called the Schur multiplier of $G$ relative to the Liezation functor or Schur Lie-multiplier. For a two-sided ideal $N$ of a Leibniz algebra $G$, we construct a four-term exact sequence relating the Schur Lie-multiplier of $G$ and $G/N$, which is applied to study and characterize Lie-nilpotency, Lie-stem covers and Lie-capability of Leibniz algebras.
We define an extension of the affine Brauer algebra, the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group and it naturally acts on $END_K(X \otimes V^{\otimes k})$ for Orthogonal and Symplectic groups. Thus we obtain a compact analogue of Schur-Weyl duality. We study functors $F_{μ,k}$ from the category of admissible $O(p,q)$ or $Sp_{2n}(\mathbb{R})$ modules to representations of the type B/C affine Brauer algebra $\mathfrak{B}_k^θ$. Thus providing a Akawaka-Suzuki-esque link between $O(p,q)$ (or $Sp_{2n}(\mathbb{R})$) and $\mathfrak{B}_k^θ$. Furthermore thes…
The paper is devoted to obtain an upper bound for the Schur multiplier of nilpotent Lie algebras of maximal class. It improves the later upper bounds on the Schur multiplier of such Lie algebras.
Recently, in [18] the authors gave some results on the structure, capability and the Schur multiplier of generalized Heisenberg Lie superalgebra. In this work we try to extend these concepts to the case of generalized Heisenberg Lie superalgebra.
In this article, we study the notion of central extension of Leibniz $n$-algebras relative to $n$-Lie algebras to study properties of Schur $\mathsf{Lie}$-multiplier and $\mathsf{Lie}$-covers on Leibniz $n$-algebras. We provide a characterization of $\mathsf{Lie}$-perfect Leibniz $n$-algebras by means of universal $\mathsf{Lie}$-central extensions. It is also provided some inequalities on the dimension of the Schur $\mathsf{Lie}$-multiplier of Leibniz $n$-algebras. Analogue to Wiegold [38] and Green [17] results on groups or Moneyhun [26] result on Lie algebras, we provide upper bounds for the…
We consider the Schur multipliers of finite dimensional nilpotent Lie algebras. If the algebra has dimension greater than one, then the Schur multiplier is non-zero. We give a direct proof of an upper bound for the dimension of the Schur multiplier as a function of class and the minimum number of generators of the algebra. We then compare this bound with another known bound.
We investigate the action of Schur algebra on the Lie algebras of derivations of free Lie algebras and operad structures constructed from it. We also show that the Lie algebra of derivations is generated by quadratic derivations together with the action of the Schur operad. Applications to certain subgroups of the automorphism group of a finitely generated free group are given as well.
By considering the nilpotent Lie algebra with the derived subalgebra of dimension $ 2$, we compute some functors include the Schur multiplier, the exterior square and the tensor square of these Lie algebras. We also give the corank of such Lie algebras.
By studying certain kind of centralizer algebras of the affine Schur algebra $\widetilde{S}(n,r)$ we show that $\widetilde{S}(n,r)$ is Noetherian and we determine its center. Assuming $n\geq r$, we show that $\widetilde{S}(n+1,r)$ is Morita equivalent to $\widetilde{S}(n,r)$, and the Schur functor is an equivalence under certain conditions.
In this paper we consider the following question: Is it possible to construct all real root representations of a given quiver Q by using universal extension functors, starting with a real Schur representation? We give a concrete example answering this question negatively.
Similar to works of G. Ellis (1998), the concept of covering pair of Lie algebras is defined. Also, we show the existence of covering pair for the pair of Lie algebras (L,N) and then show that every crossed module is a homomorphic image of a covering pair of (L,N). Finally, under some condition, we present some inequalities for the Schur multiplier of a pair of finite dimensional Lie algebras.
For a non-abelian Lie algebra $L$ of dimension $n$ with the derived subalgebra of dimension $m$ , the first author earlier proved that the dimension of its Schur multiplier is bounded by $\frac{1}{2}(n+m-2)(n-m-1)+1$. In the current work, we give some new inequalities on the exterior square and the Schur multiplier of Lie algebras and then we obtain the class of all nilpotent Lie algebras which attains the above bound. Moreover, we also improve this bound as much as possible.
From [Problem 1729, Groups of prime power order, Vol. 3], Berkovich et al. asked to obtain the Schur multiplier and the representation of a group $G$, when $G$ is a special $p$-group minimally generated by $d$ elements and $|G'|=p^{\frac{1}{2}d(d-1)}$. Since there are analogies between groups and Lie algebras, we intend to give an answer to this question similarly for nilpotent Lie algebras. Furthermore, we give some results about the tensor square and the Schur multiplier of some nilpotent Lie algebras of class two.
In the present paper, we study the notion of the Schur multiplier $\mathcal{M}(L)$ of an $n$-Lie superalgebra $L$, and prove that $\dim \mathcal{M}(L) \leq \sum_{i=0}^{n} {m\choose{i}} \mathcal{L}(n-i,k)$, where $\dim L=(m|k)$, $\mathcal{L}(0,k)=1$ and $\mathcal{L}(t,k) = \sum_{j=1}^{t}{{t-1}\choose{j-1}} {k\choose j}$, for $1\leq t\leq n$. Moreover, we obtain an upper bound for the dimension of $\mathcal{M}(L)$ in which $L$ is a nilpotent $n$-Lie superalgebra with one-dimensional derived superalgebra. It is also provided several inequalities on $\dim\mathcal{M}(L)$ as well as an $n$-Lie super…
We prove how the universal enveloping algebra constructions for Lie-Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie-Rinehart algebra morphisms, of modules over Lie-Rinehart algebras and of the infinitesimal gauge algebra of a module.
For a nilpotent Lie algebra $L$ of dimension $n$ and dim$(L^2)=m$, we find the upper bound dim$(M(L))\leq {1/2}(n+m-2)(n-m-1)+1$, where $M(L)$ denotes the Schur multiplier of $L$. In case $m=1$ the equality holds if and only if $L\cong H(1)\oplus A$, where $A$ is an abelian Lie algebra of dimension $n-3$ and H(1) is the Heisenberg algebra of dimension 3.
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 show that the centraliser of the maximal compact subgroup of the real orthogonal or symplectic groups acting on tensors of their standard representation are isomorphic to cyclotomic Brauer algebras. We also show that for the symplectic group this cyclotomic Brauer algebra splits into summands of walled Brauer algebras.
We exhibit an explicit construction for the second cohomology group $H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$. We show how the elements of $H^2(L, A)$ correspond one-to-one to the equivalence classes of central extensions of $L$ by $A$, where $A$ now is considered as an abelian Lie ring. For a finite Lie ring $L$ we also show that $H^2(L, \C^*) \cong M(L)$, where $M(L)$ denotes the Schur multiplier of $L$. These results match precisely the analogue situation in group theory.
We introduce the notion of c-nilpotent Schur Lie-multiplier of Leibniz algebras. We obtain exact sequences and formulas of the dimensions of the underlying vector spaces relating the c-nilpotent Schur Lie-multiplier of a Leibniz algebra Q and its quotient by a two-sided ideal. These tools are used to characterize Lie-nilpotency and c-Lie-stem covers of Leibniz algebras. We prove the existence of c-Lie-stem covers for finite dimensional Leibniz algebras and the non existence of c-covering on certain Lie-nilpotent Leibniz algebras with non trivial c-nilpotent Schur Lie-multiplier, and we provide…
Associated to the two types of finite dimensional simple superalgebras, there are the general linear Lie superalgebra and the queer Lie superalgebra. The universal enveloping algebras of these Lie superalgebras act on the tensor spaces of the natural representations and, thus, define certain finite dimensional quotients, the Schur superalgebras and the queer Schur superalgebra. In this paper, we introduce the quantum analogue of the queer Schur superalgebra and investigate the presentation problem for both the queer Schur superalgebra and its quantum analogue.
In this paper, we study the structure and representability of the automorphism group functor of the N=4 Lie conformal superalgebra over an algebraically closed field k of characteristic zero.
Let k be an algebraically closed field of characteristic p>0, let m,r be integers with m\ge1, r\ge0 and m\ge r and let S_0(2m,r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogon…
In this paper we define isoclinism for Lie superalgebras and using the concept of isoclinism, we give the structure of all covers of Lie superalgebras when their Schur multipliers are finite dimensional. It has been shown that that maximal stem extensions of Lie superalgebras are precisely same as the stem covers. Furthermore, we have defined stem Lie superalgebra and prove that each isoclinic family $\mathcal{C}$ contain a stem Lie superalgebra $T$ and it is the one having minimum even and odd dimension. Finally we have proved a converse of Schur's theorem and have given bound for stem Lie su…
Let $A_n(K)$ be the Kostant form of $\mathfrak{U}(sl_n^+)$ and $Γ$ the monoid generated by the positive roots of $sl_n$. For each $λ\in Λ(n,r)$ we construct a functor $F_λ$ from the category of finitely generated $Γ$-graded $A_n(K)$-modules to the category of finite dimensional $S^+(n,r)$-modules, with the property that $F_λ$ maps (minimal) projective resolutions of the one-dimensional $A_n(K)$-module $K_{A}$ to (minimal) projective resolutions of the simple $S^+(n,r)$-module $K_λ$.
The aim of this paper is to study modules over Hom-post-Lie algebras and give some contructions and various twisting i.e. we show that modules over post-Hom-Lie algebras are close by twisting either Hom-post-Lie algebras or module structure maps. Given a type of Hom-algebra A, an A-bimodule M and an O-operator T : A ---> M, we give constructions another of Hom-algebra structure on M.
In this article we investigate the interplay between stem covers, the Schur multiplier of Leibniz crossed modules and the non-abelian exterior product of Leibniz algebras. In concrete, we obtain a six-term exact sequence associated to a central extension of Leibniz crossed modules, which is useful to characterize stem covers. We show the existence of stem covers and determine the structure of all stem covers of Leibniz crossed modules. Also, we give the connection between the stem cover of a Lie crossed module in the categories of Lie and Leibniz crossed modules respectively.
We show that the hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme with coefficients in an L-module can be expressed as a derived functor. We use this fact to study a Hochschild-Serre type spectral sequence attached to an extension of Lie algebroids.
We present a novel approach to the problem of integrating homotopy Lie algebras by representing the Maurer-Cartan space functor with a universal cosimplicial object. This recovers Getzler's original functor but allows us to prove the existence of additional, previously unknown, structures and properties. Namely, we introduce a well-behaved left adjoint functor, we establish functoriality with respect to infinity-morphisms, and we construct a coherent hierarchy of higher Baker-Campbell-Hausdorff formulas. Thanks to these tools, we are able to establish the most important results of higher Lie t…
Given an algebraically closed field $\Bbbk$ of characteristic zero, a Lie superalgebra $\mathfrak{g}$ over $\Bbbk$ and an associative, commutative $\Bbbk$-algebra $A$ with unit, a Lie superalgebra of the form $\mathfrak{g} \otimes_\Bbbk A$ is known as a map superalgebra. Map superalgebras generalize important classes of Lie superalgebras, such as, loop superalgebras (where $A=\Bbbk[t, t^{-1}]$), and current superalgebras (where $A=\Bbbk[t]$). In this paper, we define Weyl functors, global and local Weyl modules for all map superalgebras where $\mathfrak{g}$ is either $\mathfrak{sl} (n,n)$ with…
Let $\mathfrak{g}$ be a reductive Lie algebra. We give a condition that ensures that the character of a generalized Verma module is well-behaved under a twisting functor. We show that a similar result holds for basic classical simple Lie superalgebras, and generalize a result from \cite{CM} about twisting Verma modules.
Recently, the authors obtained the Schur multiplier, the non-abelian tensor square and the non-abelian exterior square of $d$-generator generalized Heisenberg Lie algebras of rank $ \frac{1}{2}d(d-1).$ Here, we intend to obtain the same results for $d$-generator generalized Heisenberg Lie algebras of rank $ t$ when $ \frac{1}{2}d(d-1)-3 \leq t\leq \frac{1}{2}d(d-1)-1.$ Then, as a result, we give similar consequences for a nilpotent Lie algebra $ L $ of class two when $ \dim (L/Z(L))=d,$ $ \dim L^2=t $ such that $ \frac{1}{2}d(d-1)-3 \leq t\leq \frac{1}{2}d(d-1)-1.$
It is known that there exists a natural functor $Φ$ from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of categories, was found by Koszul [18]; this result was later extended by other authors, to different levels of generality, but always elaborating on Koszul's original idea. In this paper, I provide two new backwards equivalences, i.e. two different functors $Ψ^\circ$ and $Ψ^e$ that construct a Lie supergroup (thought of as a special group-valued functor) out of a given s…
