Ivan Losev
W-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete classification of finite dimensional irreducible modules for W-algebras. Also we study a relation between Harish-Chandra bimodules and bimodules over $W$-algebras.
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The article is devoted to some ``strange'' phenomena of representation theory and their interrelations. Cross-projective representations of pairs of anticommutative algebras, alloys, their universal envelopping Lie algebras and their representations, quaternary algebras and their alloyability are discussed. Considered examples allow to conclude that new representations have some intriguing features (continuous moduli of finite-dimensional irreducible representations, sophisticated Clebsch-Gordan coefficient calculus, etc.).
We introduce the notion of a generalized representation of a Jordan algebra with unit. The greneralized representation has the following properties: (1) Usual representations and Jacobson representations correspond to special cases of generalized representations. (2) Every simple Jordan algebra has infinitely many nonequivalent generalized representations. (3) There is a one-to-one correspondence between irreducible generalized representations of a Jordan algebra A and irreducible representations of a graded Lie algebra L(A)=U_{-1}\oplus U_0\oplus U_1 corresponding to A (the Lie algebra L(A) c…
Let $B$ be a representation-finite $\mathbb{C}$-algebra. The $\mathbb{Z}$-Lie algebra $L(B)$ associated with $B$ has been defined by Ch. Riedtmann. If $B$ is representation-directed there is another $\mathbb{Z}$-Lie algebra associated with $B$ defined by C. M. Ringel and denoted by $\CK(B)$. We prove that the Lie algebras $L(B)$ and $\CK(B)$ are isomorphic for any representation-directed $\mathbb{C}$-algebra $B$.
Let $(W,S)$ be a Coxeter system. A $W$-graph encodes a representation of the Hecke algebra $\mathcal{H}$ of $W$. We construct universal representations of multi-parameter Hecke algebras on certain quotients of path algebras, and study their relationships with $W$-graph representations. We also study the quotients of path algebras on their own, motivated by one example where the quotient path algebra is isomorphic to an ideal of Lusztig asymptotic Hecke algebra. Finally, we describe a method to obtain a generating set for the ideals by which we quotient the path algebras.
We compute two representations of COHA for $A_1$-quiver. The two untwisted representations can be combined into a representation of $D_{n+1}$ Lie algebra. The untwisted increasing representation and the twisted decreasing representation can be combined into a representation of a finite Clifford algebra.
This book treats: - spectral theory of Banach *-algebras, - basic representation theory of normed *-algebras, - spectral theory of representations of commutative *-algebras. A novel feature of the book is the construction of the enveloping C*-algebra of a general normed *-algebra.
We give a necessary and sufficient condition on parameters for Ariki-Koike algebras (resp. cyclotomic q-Schur algebras) to be of finite representation type.
In this short paper, we study $n$-representation-finite algebras from the viewpoint of the fractionally Calabi-Yau property. We shall show that all $n$-representation-finite algebras are twisted fractionally Calabi-Yau. We also show that for any $\ell>0$, twisted $\frac{n(\ell-1)}{\ell}$-Calabi-Yau algebras of global dimension at most $n$ are $n$-representation-finite. As an application, we give a construction of $n$-representation-finite algebras using the tensor product.
The tensorial description of the Auslander algebras of representation-finite string algebras
本文研究了表示有限型箭图代数的Auslander代数，通过引入粘合代数概念证明其是该代数的商，并精确描述了Dynkin型A和D两类代数的Auslander代数及其表示型。
This paper is meant to be an informative introduction to spinor representations of Clifford algebras. In this paper we will have a look at Clifford algebras and the octonion algebra. We begin the paper looking at the quaternion algebra $\mathbb{H}$ and basic properties that relate Clifford algebras and the well know Pin and Spin groups. We then will look at generalized spinor representations of Clifford algebras, along with many examples. We conclude the paper looking at the octonion algebra $\mathbb{O}$. This paper provides background to constructing representations which can be used to look …
We introduce stability categories for diagram algebras---analogues to Randal-Williams and Wahl's homogeneous categories. We use these to study representation stability properties of the Temperley--Lieb algebras, the Brauer algebras, and the partition algebras.
We show that a certain tensor space representation of the blob algebra (an affine Hecke algebra quotient) is faithful.
On minimal representation-infinite algebras
本文在代数闭域上对所有双边理想格非分配的极小表示无限代数进行了分类，并得出每个维度上此类代数的同构类仅有有限多个的结论。
We are going to show that the representation dimension of a cluster-concealed algebra B is 3. We compute its representation dimension by showing an explicit Auslander generator for the cluster-tilted algebra.
Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study cohomology and representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.
We compute the Hochschild cohomology groups of the cluster-tilted algebras of finite representation type.
We realize the current algebra of a Kac-Moody algebra as a quotient of a semi-direct product of the Kac-Moody Lie algebra and the free Lie algebra of the Kac-Moody algebra. We use this realization to study the representations of the current alg ebra. In particular we see that every ad-invariant ideal in the symmetric algebra of the Kac-Moody algebra gives rise in a canonical way to a representation of the current algebra. These representations include certain well-known families of representations of the current algebra of a simple Lie algebra. Another family of examples, which are the classic…
We investigate the fundamental properties of quantum Borcherds-Bozec algebras and their representations. Among others, we prove that the quantum Borcherds-Bozec algebras have a triangular decomposition and the category of integrable representations is semi-simple.
The aim of this work is to study the representation dimension of cluster tilted algebras. We prove that the weak representation dimension of tame cluster tilted algebras is equal to three. We construct a generator module that reaches the weak representation dimension, unfortunately this module is not always a cogenerator. We show for which algebras this module gives the representation dimension.
We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type $A$ (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic $p$. Secondly, the modular branching rules for these alg…
Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. We introduce principal series representations for these algebras. We study these representations and partially generalize Kato and Matsumoto irreducibility criteria.
In this article we describe the projective representation of Plesken Lie algebras and equivalent central extensions of these algebras. Further it is also shown that there exists a bijective correspondence between second cohomology group, equivalent central extensions and projectively equivalent projective representations of Plesken Lie algebras.
In this paper we prove that every irreducible representation of a Leibniz algebra can be obtained from irreducible representations of the semisimple Lie algebra from the Levi decomposition. We also prove that - in general - for (semi)simple Leibniz algebras it is not true that a representation can be decomposed to a direct sum of irreducible components.
We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.
Brick-finite skew-gentle algebras are representation-finite
本文证明斜有限非正则代数与表示有限性等价，扩展了Plamondon关于温和代数的结果。
In our previous work arXiv:1306.1770, we found all Borel-Schur algebra of finite representation type. In the present article, we determine which Borel-Schur algebras of infinite representation type are tame, and which are wild.
Representation-finite tensor product algebras
本文利用箭图与关系完成了表示有限张量积代数的分类，给出了完整刻画。
On representations of algebras with radical square zero
本文研究了具有零平方根理想代数的表示与物种表示之间的对应关系，并展示了如何通过物种的Auslander-Reiten quiver来重建代数的Auslander-Reiten quiver。
A bivariate representation of a complex simple Lie algebra is an irreducible representation having highest weight a combination of the first two fundamental weights. For a complex classical Lie algebra, we establish an expression for the weight multiplicities of bivariate representations.
We determine the representations of the Yokonuma-Temperley-Lieb algebra, which is defined as a quotient of the Yokonuma-Hecke algebra by generalising the construction of the classical Temperley-Lieb algebra.
In this paper we determine the complex generic representation theory of the Juyumaya algebra. We show that a certain specialization of this algebra is isomorphic to the small ramified partition algebra, introduced by P. Martin.
For monomial special multiserial algebras, which in general are of wild representation type, we construct radical embeddings into algebras of finite representation type. As a consequence, we show that the representation dimension of monomial and self-injective special multiserial algebras is less or equal to three. This implies that the finitistic dimension conjecture holds for all special multiserial algebras.
We introduce and study the so-called serpentine representations of the infinite symmetric group $\sinf$, which turn out to be closely related to the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ and representations of the Virasoro algebra. This is a new version of the manuscript of the same authors "On a relation between the basic representation of the affine Lie algebra $\widehat\sl$ and a Schur--Weyl representation of the infinite symmetric group" (arXiv:1403.1558)
We study the connection between *-representations of algebras associated with graphs, locally-scalar graph representations and the problem about the spectrum of a sum of two Hermitian operators. For algebras associated with Dynkin graphs we give an explicit description of the parameters for which there are irreducible representations and an algorithm for contructing these representations.
We study representations of the Loop Kac-Moody Lie algebra g \otimes A, where g is any Kac-Moody algebra and A is a ring of Laurent polynomials in n commuting variables. In particular, we study representations with finite dimensional weight spaces and their graded versions. When we specialize g to be a finite dimensional or affine Lie algebra we obtain modules for toroidal Lie algebras.
We construct an infinite tower of irreducible calibrated representations of periplectic Brauer algebras on which the cup-cap generators act by nonzero matrices. As representations of the symmetric group, these are exterior powers of the standard representation (i.e. hook representations). Our approach uses the recently-defined degenerate affine periplectic Brauer algebra, which plays a role similar to that of the degenerate affine Hecke algebra in representation theory of the symmetric group. We write formulas for the representing matrices in the basis of Jucys--Murphy eigenvectors and we comp…
We classify the finite dimensional irreducible representations of affine Hecke algebras of type B_2 with unequal parameters.
We prove that the representation dimension of finite dimensional selfinjective algebras over a field is invariant under socle equivalence and derive some consequences.
The representation dimension of an artin algebra as introduced by M.Auslander in his Queen Mary Notes is the minimal possible global dimension of the endomorphism ring of a generator-cogenerator. The paper is based on two texts written in 2008 in connection with a workshop at Bielefeld. The first part presents a full proof that any torsionless-finite artin algebra has representation dimension at most 3, and provides a long list of classes of algebras which are torsionless-finite. In the second part we show that the representation dimension is adjusted very well to forming tensor products of al…
Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we introduced principal series representations for these algebras and partially generalized Kato's irreducibility criterion. In this paper, we study how some of these representations decompose when they are reducible and deduce information on the irreducible representations of these algebras.
Using the skew-symmetry of the differential operators and multiplication operators in the canonical representations of finite-dimensional classical Lie algebras, we obtain some noncanonical polynomial representations of the classical Lie algebras. The representation spaces of all polynomials are decomposed into irreducible submodules, which are infinite-dimensional. Bases for the irreducible submodules are constructed. In particular, we obtain some new infinite-dimensional irreducible modules of symplectic Lie algebras that are not of highest weight type.
This is an expository article on representation theory of toroidal Lie algebras. We summerize all the results on representation theory of toroidal Lie algebras obtained in the last fifteen years. Apart from that a natural genaralization of Virasoro Lie algebra is introduced and the related results are stated. A genaralization of the semi direct product of affine Lie algebra and Virasoro Lie algebra is introduced which we call the full toroidal Lie algebra. The connection to the theory of Extended Affine Lie Algebras is explained.
We show that all finite dimensional, tame hereditary $k$-algebras are of amenable representation type (in the sense of G. Elek) for all fields $k$. The proof is adapted from our previous result for tame path algebras. Further, it is proven that this results extends to some tilted algebras, in particular tame concealed algebras are amenable.
Representations of compatible Lie algebras
从代数和表示论角度研究相容李代数，给出经典李代数理论中李定理、外尔定理和列维定理的反例，并分类二维相容李代数及其表示。
We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted algebras of finite representation type are obtained along the way.
We initiate the representation theory of the degenerate affine periplectic Brauer algebra on $n$ strands by constructing its finite-dimensional calibrated representations when $n=2$. We show that any such representation that is indecomposable and does not factor through a representation of the degenerate affine Hecke algebra occurs as an extension of two semisimple representations with one-dimensional composition factors; and furthermore, we classify such representations with regular eigenvalues up to isomorphism.
We start the study of glider representations in the setting of semisimple Lie algebras. A glider representation is defined for some positively filtered ring $FR$ and here we consider the right bounded algebra filtration $FU(\mathfrak{g})$ on the universal enveloping algebra $U(\mathfrak{g})$ of some semisimple Lie algebra $\mathfrak{g}$ given by a fixed chain of semisimple sub Lie algebras $\mathfrak{g}_1 \subset \mathfrak{g}_2 \subset \ldots \subset \mathfrak{g}_n = \mathfrak{g}$. Inspired by the classical representation theory, we introduce so-called Verma glider representations. Their exist…
We consider the representation dimension, for fixed $n\geq2$, of ordinary and quantised Schur algebras $S(n,r)$ over a field $k$. For $k$ of positive characteristic $p$ we give a lower bound valid for all $p$. We also give an upper bound in the quantum case, when $k$ has characteristic $0$.
We survey some important results concerning the finite--dimensional representations of the loop algebra of a simple complex Lie algebra, and their twisted loop subalgebras. In particular, we review the parametrization and description of the Weyl modules and of the irreducible finite--dimensional representations of such algebras, describe a block decomposition of the (non--semisimple) category of their finite--dimensional representations, and conclude with recent developments in the representation theory of multiloop algebras.
To a generic holomorphic vector bundle on an algebraic curve and an irreducible finite-dimensional representation of a semisimple Lie algebra, we assign a representation of the corresponding affine Krichever--Novikov algebra in the space of semi-infinite exterior forms. It is shown that equivalent pairs of data give rise to equivalent representations and vice versa.
An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras.
This book treats: - spectral theory of Banach *-algebras, - basic representation theory of normed *-algebras, - spectral theory of representations of commutative *-algebras. A novel feature of the book is the construction of the enveloping C*-algebra of a general normed *-algebra.
We establish bounds for the representation dimension of skew group algebras and wreath products. Using this, we obtain bounds for the representation dimension of a block of a Hecke algebra of type A, in terms of the weight of the block. This includes certain blocks of group algebras of symmetric groups.
On BD-algebra and CM-Auslander algebra for a gentle algebra and their representation types
本文研究了gentle代数及其BD-gentle代数与CM-Auslander代数，证明了这三类代数的表示有限性与表示离散性均保持一致。
Classification of the d-representation-finite symmetric k-algebras of finite representation type
本文基于d-表示有限对称Nakayama代数和路径代数的平凡延伸，对有限表示类型的d-表示有限对称代数进行分类。
We construct some irreducible representations of the Leavitt path algebra of an arbitrary quiver. The constructed representations are associated to certain algebraic branching systems. For a row-finite quiver, we classify algebraic branching systems, to which irreducible representations of the Leavitt path algebra are associated. For a certain quiver, we obtain a faithful completely reducible representation of the Leavitt path algebra. The twisted representations of the constructed ones under the scaling action are studied.
Let k be a field. A finite dimensional k-algebra is said to be minimal representation-infinite provided it is representation-infinite and all its proper factor algebras are representation-finite. Our aim is to classify the special biserial algebras which are minimal representation-infinite. The second part describes the corresponding module categories.
We use evaluation representations to give a complete classification of the finite-dimensional simple modules of twisted current algebras. This generalizes and unifies recent work on multiloop algebras, current algebras, equivariant map algebras, and twisted forms.
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables.
It is shown that the derived dimension of any representation-finite Artin algebra is at most one.
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