We calculate Ext^*_{SL_2(k)}(Δ(λ), Δ(μ)), Ext^*_{SL_2(k)}(L(λ), Δ(μ)), Ext^*_{SL_2(k)}(Δ(λ), L(μ)), and Ext^*_{SL_2(k)}(L(λ), L(μ)), where Δ(λ) is the Weyl module of highest weight λ, L(λ) is the simple SL_2(k)-module of highest weight λand our field k is algebraically closed of positive characteristic. We also get analogous results for the Dipper-Donkin quantisation. To do thus we construct the Lyndon-Hochschild-Serre spectral sequence in a new way, and find a new condition for the E_2 page of any spectral sequence to be the same as the E_\infty page.
In this paper, we describe the Galois module structure of $J=\mathbf{K}^{\times}/\mathbf{K}^{\times p}$, where $\mathbf{K}$ is an extension of a local field $\mathbf{k}$ containing a primitive $p$-th root of unity: for instance, if $\mathbf{K}/\mathbf{k}$ is a $p$-elementary abelian extension, we prove that $J$ is a module of constant Jordan type, with stable Jordan type $[1]^2$, which, in a way, extends the result of J. Mináč and J. Swallow. Also, we take profit from our proof by computing some invariants, which were previously introduced by A. Adem, W. Gao, D. B. Karageuzian and J. Mináč onl…
In this article we are examining extensions and some basic diagrammatic properties of modules, in both cases from a new, "virtual" point of view. As natural background for investigating the kind of problems we are dealing with, the virtual category of a module M is introduced, having as objects the submodules of M's subquotients modulo some identifications. In the case of extensions our approach implies viewing "proportionality classes" of extensions of (dually, by) a simple module by (resp. of) another simple as quotients of a certain quotient (which is in fact a subdirect product) of a proje…
Let $A$ be the one point extension of an algebra $B$ by a projective $B$-module. We prove that the extension of a given support $τ$-tilting $B$-module is a support $τ$-tilting $A$-module; and, conversely, the restriction of a given support $τ$-tilting $A$-module is a support $τ$-tilting $B$-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support $τ$-tilting modules.
Computing the extensions between Verma modules is in general a very difficult problem. Using Soergel bimodules, one can construct a graded version of the principal block of Category $\mathcal{O}$ for any finite coxeter group. In this setting, we compute the extensions between Verma modules for dihedral groups.
New exact modular branching rules are obtained for modules over the symmetric groups that are close to completely splittable modules. These results are based on some upper bounds for the Ext^1-spaces between simple modules.
Let kK be the path algebra of the Kronecker quiver and consider the category of finite dimensional right modules over kK (called Kronecker modules). We prove that extensions of Kronecker modules are field independent up to Segre classes, so they can be described purely combinatorially. We use in the proof explicit descriptions of particular extensions and a variant of the well known Green formula for Ringel-Hall numbers, valid over arbitrary fields. We end the paper with some results on extensions of preinjective Kronecker modules, involving the dominance ordering from partition combinatorics …
Let $B$ be the one-point extension algebra of $A$ by an $A$-module $X$. We proved that every support $τ$-tilting $A$-module can be extended to be a support $τ$-tilting $B$-module by two different ways. As a consequence, it is shown that there is an inequality $$|\stilt B|\geqslant 2|\stilt A|.$$
We provide a classification of generalized tilting modules and full exceptional sequences for the dual extension algebra of the path algebra of a uniformly oriented linear quiver modulo the ideal generated by paths of length two with its opposite algebra. For the classification of generalized tilting modules we develop a combinatorial model for the poset of indecomposable self-orthogonal modules with standard filtration with respect to the relation arising from higher extensions.
We prove that the categories of Gelfand-Zeitlin modules of $\mathfrak{g}=\mathfrak{gl}_n$ and Whittaker modules associated with a semi-simple complex finite-dimensional algebra $\mathfrak{g}$ are extension full in the category of all $\mathfrak{g}$-modules. This is used to estimate and in some cases determine the global dimension of blocks of the categories of Gelfand-Zeitlin and Whittaker modules.
We describe how the center of the universal central extension of the genus zero Krichever-Novikov current algebra decomposes as a direct sum of irreducible modules for automorphism group of the coordinate ring of this algebra.
In this paper, we establish an explicit classification of length two extensions of tensor modules for the Witt algebra using the cohomology of the Witt algebra with coefficients in the module of the space of homomorphisms between the two modules of interest. To do this we extended our module to a module that has a compatible action of the commutative algebra of Laurent polynomials in one variable. In this setting, we are be able to directly compute all possible 1-cocycles.
We introduce the notion of totally reflexive extension of rings. It unifies Gorenstein orders and Frobenius extensions. We prove that for a totally reflexive extension, a module over the extension ring is totally reflexive if and only if its underlying module over the base ring is totally reflexive.
A conjecture going back to the eighties claims that there are no non-trivial self-extensions of irreducible modules over symmetric groups if the characteristic of the ground field is not equal to $2$. We obtain some partial positive results on this conjecture.
Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand. Secondly, we discuss the behaviour of minimality under ring extensions. We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism. Similar re…
Let $B$ be an one-point extension of a finite dimensional $k$-algebra $A$ by a simple $A$-module at a source point $i$. In this paper, we classify the $τ$-tilting modules over $B$. Moreover, it is shown that there are equations $$|\tilt B|=|\tilt A|+|\tilt A/\langle e_i\rangle|\quad \text{and}\quad |\stilt B|=2|\stilt A|+|\stilt A/\langle e_i\rangle|.$$ As a consequence, we can calculate the numbers of $τ$-tilting modules and support $τ$-tilting modules over linearly Dynkin type algebras whose square radical are zero.
We prove that the monoid of generic extensions of finite dimensional nilpotent $k[T]$-modules is isomorphic to the monoid of partitions (with addition of partitions). Moreover we give a combinatorial algorithm that calculates constant terms of classical Hall polynomials.
Degeneration of modules is usually defined geometrically, but due to results of Zwara and Riedtmann we can also define it in terms of exact sequences. This definition also works over fields that are not algebraically closed. Let $k$ be a field, $K$ a finite extension of $k$ and $Λ$ a $k$-algebra. Then any $K\otimes_kΛ$-module is also a $Λ$-module. We study how the isomorphism classes, degeneration and hom-order differ depending on which algebra we are working over.
The irreducible representations of the Witt algebra $W$ are completely known. A classification of the irreducible $U_χ(W)$--modules was first established by Chang and later simplified by Strade. The aim of this article is to give a classification of the extensions of the simple $U_χ(W)$--modules having $p$--character $χ$ of height $-1,0,1$ and $p-1$ where $p$ denotes the characteristic of the ground field.
A recent result of N. Abe implies that the Gabber-Joseph conjecture is true for the first-degree extensions between Verma modules with regular integral highest weights.
In this paper, we investigate the tensor structure of the category of finite dimensional weight modules over the Hopf-Ore extensions $kG(χ^{-1}, a, 0)$ of group algebras $kG$. The tensor product decomposition rules for all indecomposable weight modules are explicitly given under the assumptions that $k$ is an algebraically closed field of characteristic zero, and the orders of $χ$ and $χ(a)$ are the same.
Twisted current algebras are fixed point subalgebras of current algebras under a finite group action. Special cases include equivariant map algebras and twisted forms of current algebras. Their finite-dimensional simple modules fall into two categories, those which factor through an evaluation map and those which do not. We show that there are no nontrivial extensions between finite-dimensional simple evaluation and non-evaluation modules. We then compute extensions between any pair of finite-dimensional simple modules for twisted current algebras, and use this information to determine the blo…
We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete classification of the projective dimension of a C-module inside the module category of B. If a C-module M is rigid, we show two sufficient conditions for M to be a rigid B-module. In particular, if M is an indecomposable and rigid C-module, we prove M is always a rigid B-module.
We calculate the first extension groups for finite-dimensional simple modules over an arbitrary generalized current Lie algebra, which includes the case of loop Lie algebras and their multivariable analogs.
Let $Γ$ be a group acting on a scheme $X$ and on a Lie superalgebra $\mathfrak{g}$, both defined over an algebraically closed field of characteristic zero $\Bbbk$. The corresponding equivariant map superalgebra $M(\mathfrak{g}, X)^Γ$ is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak{g}$. In this paper we complete the classification of finite-dimensional irreducible $M(\mathfrak{g}, X)^Γ$-modules when $\mathfrak{g}$ is a finite-dimensional simple Lie superalgebra, $X$ is of finite type and $Γ$ is a finite abelian group acting freely on the rational points of $X$, by cla…
We study self-extensions of modules over symmetric artin algebras. We show that non-projective modules with eventually vanishing self-extensions must lie in AR components of stable type $\mathbb{Z}A_{\infty}$. Moreover, the degree of the highest non-vanishing self-extension of these modules is determined by their quasilength. This has implications for the Auslander-Reiten Conjecture and the Extension Conjecture.
Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we give a complete classification of extensions of finite irreducible conformal modules of $\mathcal{W}(a,b)$. With a similar method, we characterize all extensions of finite irreducible conformal modules of Schrödinger-Virasoro type Lie conformal algebras $TSV(a,b)$ and $TSV(c)$.
This paper concerns representations of the integral general linear group. The extension groups $Ext^2$ between any pair of hook Weyl modules are determined via a detailed study of cyclic generators and relations associated to certain extensions. As a corollary, the modular extension groups $Ext^1$ between such modules are determined.
Let p be a prime number. We compute the Yoneda extension algebra of $GL_2$ over an algebraically closed field of characteristic p by developing a theory of Koszul duality for a certain class of 2-functors, one of which controls the category of rational representations of $GL_2$ over such a field.
We determine the integral extension groups $Ext^1(Δ(h),Δ(h(k)))$ and $Ext^k(Δ(h),Δ(h(k)))$, where $Δ(h),Δ(h(k))$ are the Weyl modules of the general linear group $GL_n$ corresponding to the hook partitions $h=(a,1^b)$, $h(k)=(a+k,1^{b-k})$.
In this paper we study extensions between Cohen-Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We prove that rank 1 modules are periodic, and we give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine ${\rm Ext}^i(L_I, L_J)$ for arbitrary rank 1 modules $L_I$ and $L_J$. An explicit combinatorial algorithm is given for computation of ${\rm Ext}^i(L_I, L_J)$ when $i$ is odd, and for $i$ even, we show that ${\rm Ext}^i(L_I, L_J)$ is cyclic over the centre, and we give an expli…
We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version of) the extension algebra of standard modules. Examples of such algebras include, in particular, the multiplicity free blocks of the BGG category $\mathcal{O}$, and some quasi-hereditary algebras with Cartan decomposition in the sense of K{ö}nig.
We study IE-closed subcategories of a module category, subcategories which are closed under taking Images and Extensions. We investigate the relation between IE-closed subcategories and torsion pairs, and characterize $τ$-tilting finite algebras using IE-closed subcategories. For the hereditary case, we show that IE-closed subcategories can be classified by twin rigid modules, pairs of rigid modules satisfying some homological conditions. Moreover, we introduce mutation of twin rigid modules analogously to tilting modules, which gives a way to calculate all twin rigid modules for the represent…
Let $\mathcal{G}$ be the planar Galilean conformal algebra and $\widetilde{\mathcal{G}}$ be its universal central extension. Then $\mathcal{G}$ (resp. $\widetilde{\mathcal{G}}$) admits a triangular decomposition: $\mathcal{G}=\mathcal{G}^{+}\oplus\mathcal{G}^{0}\oplus\mathcal{G}^{-}$ (resp. $\widetilde{\mathcal{G}}=\widetilde{\mathcal{G}}^{+}\oplus\widetilde{\mathcal{G}}^{0}\oplus\widetilde{\mathcal{G}}^{-}$). In this paper, we study universal and generic Whittaker $\mathcal{G}$-modules (resp. $\widetilde{\mathcal{G}}$-modules) of type $ϕ$, where $ϕ:\mathcal{G}^{+}=\widetilde{\mathcal{G}}^{+}\…
Let $A$ be a finite dimensional $k$-algebra standardly stratified for a partial order $\leqslant$ and $Δ$ be the direct sum of all standard modules. In this paper we study the extension algebra $E= \text{Ext}_A^{\ast} (Δ, Δ)$ of standard modules, characterize the stratification property of $E$ for $\leqslant$ and $\leqslant ^{op}$, and obtain a sufficient condition for $E$ to be a generalized Koszul algebra (in a sense which we define).
We investigate when the categories of all rational $A$-modules and of finite dimensional rational modules are closed under extensions inside the category of $C^*$-modules, where $C^*$ is the cofinite topological completion of $A$. We give a complete characterization of these two properties, in terms of a topological and a homological condition. We also give connections to other important notions in coalgebra theory such as coreflexive coalgebras. In particular, we are able to generalize many previously known partial results and answer some questions in this direction, and obtain large classes …
Ce mémoire traite de la théorie des représentations d'une certaine classe d'algèbres de Lie de dimension infinie, les algèbres de courants tordues. L'objet du travail est d'obtenir une classification des blocs d'extensions d'une catégorie de modules de dimension finie pour une algèbre de courants tordue donnée. Les principales sources de cette étude sont les récentes classifications des modules simples de dimension finie pour ces algèbres et des blocs d'extensions pour les modules de dimension finie dans le cas des algèbres d'applications équivariantes. Ces algèbres de courants tordues compren…
We exhibit an isomorphism of associative algebras between the $\operatorname{Ext}$-algebra $\operatorname{Ext}_Λ^\ast(Δ,Δ)$ of standard modules over the dual extension algebra $Λ$ of two directed algebras $B$ and $A$ and the dual extension algebra of the $\operatorname{Ext}$-algebra $\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$ with $A$. There are natural $A_\infty$-structures on these $\operatorname{Ext}$-algebras, and, under certain technical assumptions on $B$, we describe that on $\operatorname{Ext}_Λ^\ast(Δ,Δ)$ completely in terms of that on $\operatorname{Ext}_B^\ast(\mathbb{L},\mat…
Let $Σ_d$ denote the symmetric group of degree $d$ and let $K$ be a field of positive characteristic $p$. For $p>2$ we give an explicit description of the first cohomology group $H^1(Σ_d,{\rm{Sp}}(λ))$, of the Specht module ${\rm{Sp}}(λ)$ over $K$, labelled by a partition $λ$ of $d$. We also give a sufficient condition for the cohomology to be non-zero for $p=2$ and we find a lower bound for the dimension. Our method is to proceed by comparison with the cohomology for the general linear group $G(n)$ over $K$ and then to reduce to the calculation of ${\rm{Ext}}^1_{B(n)}(S^d E,K_λ)$, where $B(n)…
We construct twisted $\mathcal{D}$-modules on the projective line $\mathbb{P}^1$ that are equivariant for the action of the diagonal torus subgroup of $SL_2$. In the most interesting case these arise as extensions from local systems on $\mathbb{C}^{\times}$. We discuss their subquotient structure. Their sheaf cohomology groups are weight modules for the Lie algebra $\mathfrak{sl}_2$. We also discuss their subquotient structure and in case these modules are not the familiar highest or lowest weight modules, we give an explicit presentation for them. Our computations illustrate some basic $\math…
We introduce a family of rank-one local systems in the category of twisted $\mathcal{D}$-modules on a certain subvariety isomorphic to ${\mathbb{G}_{\text{m}}}^2$ of the affine flag variety of $\text{SL}_2$. We then give a criterion for these local systems, in terms of their parameters, to extend cleanly in the sense of $\mathcal{D}$-modules.
We study certain filtrations of indecomposable injective modules over classical Lie superalgebras, applying a general approach for noetherian rings developed by Brown, Jategaonkar, Lenagan, and Warfield. To indicate the consequences of our analysis, suppose that $g$ is a complex classical simple Lie superalgebra and that $E$ is an indecomposable injective $g$-module with nonzero (and so necessarily simple) socle $L$. (Recall that every essential extension of $L$, and in particular every nonsplit extension of $L$ by a simple module, can be formed from $g$-subfactors of $E$.) A direct transposit…
We give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.
The higher APR tilting modules and higher BB tilting modules were introduced and studied in higher Auslander-Reiten theory. Our objective is to consider these tilting modules by the corresponding simple modules, and show that the tensor product of higher APR (BB) tilting modules is a higher APR (BB) tilting module.
In this paper, we calculate the space $Ext^1_{GL(n)}(L_n(λ),L_n(μ))$, where GL(n) is the general linear group of degree $n$ over an algebraically closed field of positive characteristic, $L_n(λ)$ and $L_n(μ)$ are rational irreducible GL(n)-modules with highest weights $λ$ and $μ$ respectively, the restriction of $L_n(λ)$ to any Levi subgroup of GL(n) is semisimple, $λ$ is a $p$-restricted weight and $μ$ does not strictly dominate $λ$.
We introduce a notion of $n$-Lie Rinehart algebras as a generalization of Lie Rinehart algebras to $n$-ary case. This notion is also an algebraic analogue of $n$-Lie algebroids. We develop representation theory and describe a cohomology complex of $n$-Lie Rinehart algebras. Furthermore, we investigate extension theory of $n$-Lie Rinehart algebras by means of $2$-cocycles. Finally, we introduce crossed modules of $n$-Lie Rinehart algebras to gain a better understanding of their third dimensional cohomology groups.
