We give a complete classification of all $d$-representation-finite symmetric Nakayama algebras and of all $d$-representation-finite trivial extensions of path algebras of quivers, over an arbitrary field. As a consequence we get a classification, up to Morita equivalence, of all $d$-representation-finite symmetric algebras of finite representation type over an algebraically closed field.
We classify all graded compatible left-symmetric algebraic structures on high rank Witt algebras, and classify all non-graded ones satisfying a minor condition. Furthermore, graded compatible left-symmetric algebraic structures on high rank Virasoro algebras are also classified.
Gendo-symmetric algebras were introduced by Fang and Koenig as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost $ν$-stable derived equivalences, introduced by Hu and Xi, between representation-finite gendo-sy…
In this paper, under some natural condition, a complete classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra $\mathcal{W}(a,b)$ is presented. Moreover, applying this result, we obtain a class of compatible left-symmetric algebraic structures on the coefficient algebra of $\mathcal{W}(a,b)$.
In this paper, we establish the Composition-Diamond lemma for right-symmetric algebras. As an application, we give a Gröbner-Shirshov basis for universal enveloping right--symmetric algebra of a Lie algebra.
We study symmetric algebras $A$ over an algebraically closed field $F$ in which the Jacobson radical of the center of $A$, the socle of the center of $A$ or the Reynolds ideal of $A$ are ideals.
Let A be a finite dimensional symmetric cllular algebras. We construct a nilpotent ideal in A. The ideal connects the radicals of cell modules with the radical of the algebra. It also reveals some information on the dimensions of simple modules of A.
This paper studies self-injective algebras of polynomial growth. We prove that the derived equivalence classification of weakly symmetric algebras of domestic type coincides with the classification up to stable equivalences (of Morita type). As for weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between weakly symmetric algebras o…
We establish a lower bound for the representation dimension of all the classical Hecke algebras of types A, B and D. For all the type A algebras, and most of the algebras of types B and D, we also establish upper bounds. Moreover, we establish bounds for the representation dimension of group algebras of some symmetric groups.
The notion of silting mutation was introduced by Iyama and the author. In this paper we mainly study silting mutation for self-injective algebras and prove that any representation-finite symmetric algebra is tilting-connected. Moreover we give some sufficient conditions for a Bongartz-type Lemma to hold for silting objects.
The aim of this paper is to introduce and study quadratic Hom-Lie algebras, which are Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the double extension theory to Hom-Lie algebras. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. We establish a correspondence between the class of involutive quadratic Hom-Lie algebras and quadratic simple Lie algebras with symmetric involution. Centerless involutive quadratic Hom-Lie algebras are characterized. Also el…
In this paper, we shall recall and arrange the relationship between hyperbolic elements and parabolic subalgebras at first. And then, we shall classify all the compatible parabolic subalgebras containing a $τ$-stable Borel subalgebra for complex symmetric pairs $(\mathfrak{g},\mathfrak{g}^τ)$, where $τ$ is a involutive automorphism of $\mathfrak{g}$.
The notion of mutation plays crucial roles in representation theory of algebras. Two kinds of mutation are well-known: tilting/silting mutation and quiver-mutation. In this paper, we focus on tilting mutation for symmetric algebras. Introducing mutation of SB quivers, we explicitly give a combinatorial description of tilting mutation of symmetric special biserial algebras. As an application, we generalize Rickard's star theorem. We also introduce flip of Brauer graphs and apply our results to Brauer graph algebras.
Let $L$ be one of the finite dimensional Lie algebras $W_n({\bf m}),$ $S_n({\bf m}),$ $ H_n({\bf m})$ of Cartan type over an algebraically closed field of prime characteristic $p>0.$ For an elements $F$ of the symmetrical algebra $S(L)$ we found necessary and sufficient condition in order to the element $ad(\partial_1)^{p^{m_1}-1} ad(\partial_2)^{p^{m_2}-1}... ad(\partial_n)^{p^{m_n}-1}(F)$ belongs to the symmetrical invariants algebra $S(L)^L.$ Also, for $p=3,5$ the algebra of symmetrical invariants $S(H_2)^{H_2}$ is calculated in explicit way.
We investigate symmetric quotient algebras of symmetric algebras, with an emphasis on finite group algebras over a complete discrete valuation ring ${\mathcal O}$. Using elementary methods, we show that if an ordinary irreducible character $χ$ of a finite group $G$ gives rise to a symmetric quotient over ${\mathcal O}$ which is not a matrix algebra, then the decomposition numbers of the row labelled by $χ$ are all divisible by the characteristic $p$ of the residue field of ${\mathcal O}$.
We complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence.
As a step towards the structure theory of Lie algebras in symmetric monoidal categories we establish results involving the Killing form. The proper categorical setting for discussing these issues are symmetric ribbon categories.
In this paper we give a new definition of symmetric special multiserial algebras in terms of defining cycles. As a consequence, we show that every special multiserial algebra is a quotient of a symmetric special multiserial algebra.
We introduce the class of generalized biserial quiver algebras and prove that they provide a complete classification of all weakly symmetric biserial algebras over an algebraically closed field.
We identify a collection of primitive elements generating the Hopf algebra NCSym of symmetric functions in noncommuting variables and give a combinatorial formula for the antipode.
We provide a complete classification of all tame and wild symmetric special multiserial algebras in terms of the underlying Brauer configuration. Our classification contains the symmetric special multiserial algebras of finite representation type.
If A is a finite-dimensional symmetric algebra, then it is well-known that the only silting complexes in $\mathrm{K^b}(\mathrm{proj}A)$ are the tilting complexes. In this note we investigate to what extent the same can be said for weakly symmetric algebras. On one hand, we show that this holds for all tilting-discrete weakly symmetric algebras. In particular, a tilting-discrete weakly symmetric algebra is also silting-discrete. On the other hand, we also construct an example of a weakly symmetric algebra with silting complexes that are not tilting.
In this paper, we build the unfolding approach from acyclic sign-skew-symmetric matrices of finite rank to skew-symmetric matrices of infinite rank, which can be regard as an improvement of that in the skew-symmetrizable case. Using this approach, we give a positive answer to the problem by Berenstein, Fomin and Zelevinsky in \cite{fz3} which asks whether an acyclic sign-skew-symmetric matrix is always totally sign-skew-symmetric. As applications, the positivity for cluster algebras in acyclic sign-skew-symmetric case is given; further, the $F$-polynomials of cluster algebras are proved to hav…
We show how to extend the theory of generalized Reynolds ideals, as introduced by B. K"ulshammer, from symmetric algebras to arbitrary finite-dimensional algebras (in positive characteristic). This provides new invariants of the derived module categories of finite-dimensional algebras, extending recent results by the third author for symmetric algebras.
Motivated by the relation between Schur algebra and the group algebra of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang and Steffen Koenig addressed some behaviour of the endomorphism algebra of a generator over a symmetric algebra, which they called gendo-symmetric algebra. Continuing this line of works, we classify in this article the representation-finite gendo-symmetric algebras that have at most one isomorphism class of indecomposable non-injective projective module. We also determine their almost ν-stable derived equivalence classes in the sense of…
Gendo-symmetric algebras were recently introduced by Fang and König. An algebra is called gendo-symmetric in case it is isomorphic to the endomorphism ring of a generator over a finite dimensional symmetric algebra. We show that a finite dimensional algebra $A$ over a field $K$ is gendo-symmetric if and only if there is a bocs-structure on $(A,D(A))$, where $D=Hom_K(-,K)$ is the natural duality. Assuming that $A$ is gendo-symmetric, we show that the module category of the bocs $(A,D(A))$ is isomorphic to the module category of the algebra $eAe$, when $e$ is an idempotent such that $eA$ is the …
We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to congruence, local commutative associative algebras with zero cube radical and square radical of dimension 3, and Lie algebras with central commutator subalgebra of dimension 3 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.
We give a classification of the triples (g,g',q) such that Zuckerman's derived functor (g,K)-module A_q(λ) for a θ-stable parabolic subalgebra q is discretely decomposable with respect to a reductive symmetric pair (g,g'). The proof is based on the criterion for discretely decomposable restrictions by the first author and on Berger's classification of reductive symmetric pairs.
Extending results of Rais-Tauvel, Macedo-Savage, and Arakawa-Premet, we prove that under mild restrictions on the Lie algebra $\mathfrak q$ having the polynomial ring of symmetric invariants, the m-th Takiff algebra of $\mathfrak q$, $\mathfrak q\langle m\rangle$, also has a polynomial ring of symmetric invariants.
The classification of thick representations of the symmetric group
Center for Medical Education and Sciences, Faculty of Medicine, University of Yamanashi（山梨大学医学部医学教育与科学中心）；National Institute of Technology, Kagawa College（国立高知工业大学）；National Institute of Technology, Akashi College（国立明石工业大学）
We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.
We give a characterization of the sets of objects of the derived category of a block of a finite group algebra (or other symmetric algebra) that occur as the set of images of simple modules under an equivalence of derived categories. We give some applications to proving that certain blocks have equivalent derived categories.
We give a complete derived equivalence classification of all symmetric algebras of domestic representation type over an algebraically closed field. This completes previous work by R. Bocian and the authors, where in this paper we solve the crucial problem of distinguishing standard and nonstandard algebras up to derived equivalence. Our main tool are generalized Reynolds ideals, introduced by B. K"ulshammer for symmetric algebras in positive characteristic, and recently shown by A. Zimmermann (math.RA/0509580) to be invariants under derived equivalences.
The author classifies Klein four symmetric pairs of holomorphic type for the non-compact Lie group of Hermitian type $\mathrm{E}_{7(-25)}$, and applies the results to branching rules.
In this paper, we compute the number of two-term tilting complexes for an arbitrary symmetric algebra with radical cube zero over an algebraically closed field. Firstly, we give a complete list of symmetric algebras with radical cube zero having only finitely many isomorphism classes of two-term tilting complexes in terms of their associated graphs. Secondly, we enumerate the number of two-term tilting complexes for each case in the list.
We prove that a finite dimensional algebra $A$ with representation-finite subcategory consisting of modules that are semi-Gorenstein-projective and $n$-th syzygy modules is left weakly Gorenstein. This generalises a theorem of Ringel and Zhang who proved the result in the case $n=1$. As an application we show that monomial algebras and endomorphism rings of modules over representation-finite algebras are weakly Gorenstein. We then give a new connection between the theory of dominant dimension and Gorenstein homological algebra for gendo-symmetric algebras. As an application, we will see that t…
We show how any finite-dimensional algebra can be realized as an idempotent subquotient of some symmetric quasi-hereditary algebra. In the special case of rigid symmetric algebras we show that they can be realized as centralizer subalgebras of symmetric quasi-hereditary algebras. We also show that the infinite-dimensional symmetric quasi-hereditary algebras we construct admit quasi-hereditary structure with respect to two opposite orders, that they have strong exact Borel and $Δ$-subalgebras and the corresponding triangular decompositions.
We introduce Brauer complex of symmetric SB-algebra, and reformulate in terms of Brauer complex the so far known invariants of stable and derived equivalence of symmetric SB-algebras. In particular, the genus of Brauer complex turns out to be invariant under derived equivalence. We study transformations of Brauer complexes which preserve class of derived equivalence. Additionally, we establish a new invariant of derived equivalence of symmetric SB-algebras. As a consequence, symmetric SB-algebras with Brauer complex of genus 0 are classified. Keywords: Brauer tree algebras, special biserial al…
This is an expository article. We survey some fundamental trends in representation theory of symmetric groups and related objects which became apparent in the last fifteen years. The emphasis is on connections with Lie theory via categorification. We present results on branching rules and crystal graphs, decomposition numbers and canonical bases, graded representation theory, connections with cyclotomic and affine Hecke algebras, Khovanov-Lauda-Rouquier algebras, category ${\mathcal O}$, $W$-algebras, ...
We introduce and study symmetric and exterior algebras in braided monoidal categories such as the category O for quantum groups. We relate our braided symmetric algebras and braided exterior algebas with their classical counterparts.
We prove the classification of homomorphisms from the algebra of symmetric functions to $\mathbb{R}$ with non-negative values on Macdonald symmetric functions $P_λ$, that was conjectured by S.V. Kerov in 1992.
We develop the homological theory of KLR algebras of symmetric affine type. For each PBW basis, a family of standard modules is constructed which categorifies the PBW basis.
We show that in characteristic zero all irreducible symmetrically braided Hopf algebras are of PBW type. Consequently, we obtain conditions for a braided Hopf algebra to be of PBW type as module over a braided Hopf subalgebra containing the coradical.
We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute brackets also in some cases for skew group algebras arising as group extensions of quantum symmetric algebras.
We give a complete classification of irreducible symmetric spaces for which there exist proper SL(2,R)-actions as isometries, using the criterion for proper actions by T. Kobayashi [Math. Ann. '89] and combinatorial techniques of nilpotent orbits. In particular, we classify irreducible symmetric spaces that admit surface groups as discontinuous groups, combining this with Benoist's theorem [Ann. Math. '96].
A classic result of representation theory is Brauer's construction of a diagrammatical (geometrical) algebra whose matrix representation is a certain given matrix algebra, which is the commutating algebra of the enveloping algebra of the representation of the orthogonal group. The purpose of this paper is to provide a motivation for this result through the categorial notion of symmetric self-adjunction.
The symmetric coinvariant algebra $C[x_1, dots, x_n]_{S_n}$ is the quotient algebra of the polynomial ring by the ideal generated by symmetric polynomials vanishing at the origin. It is known that the algebra is isomorphic to the regular representation of $S_n$. Replacing $C[x]$ with $A = C[x,y]/(xy)$, we introduce another symmetric coinvariant algebra $A^{otimes n}_{S_n}$ and determine its $S_n$-module structure. As an application, we determine the $sl_{r+1}$-module structure of the local Weyl module at a double point for $sl_{r+1} otimes A$.
We consider a natural generalisation of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the algebras of [Bocian-Holm-Skowroński, J. Pure Appl. Algebra 2004], where they study the weakly symmetric algebras of Euclidean type, as well as some algebras of dihedral type.
In the previous paper "Symmetric Crystals and Affine Hecke Algebras of Type B", we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for $\gl_\infty$. In the first half of this paper (sections 2 and 3), we give a survey of the LLTA type theorem of the affine Hecke algebra of type $A$. In the latter half (sections 4, 5 and 6), we review the construction of the symmetric crystals and the LLTA type conjectures for the affine Hecke algebra of type $B$.
We study the representation theory of three towers of algebras which are related to the symmetric groups and their Hecke algebras. The first one is constructed as the algebras generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. The two others are the monoid algebras of nondecreasing functions and nondecreasing parking functions. For these three towers, we describe the structure of simple and indecomposable projective modules, together with the Cartan map. The Grothendieck algebras and coalgebras given respectively by the induct…
By computing Frobenius-Schur indicators of modules of certain weak Hopf algebras, we give a formula for the number of involutions in symmetric groups, which are contained in a given coset with respect to a given Young subgroup.
