Classification of Loop Modules with finite dimensional weight spaces

1996 ◽  
Vol 305 (1) ◽  
pp. 651-663 ◽  
Author(s):  
S. Eswara Rao
Keyword(s):  
Finite Dimensional ◽  
Loop Modules

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

scholarly journals The Lattice of Definable Equivalence Relations in Homogeneous $n$-Dimensional Permutation Structures

10.37236/5980 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Samuel Braunfeld
Keyword(s):  
Distributive Lattice ◽  
Equivalence Relations ◽  
Electronic Journal ◽  
Finite Distributive Lattice ◽  
Linear Orders ◽  
Finite Dimensional ◽  
Homogeneous Structures ◽  
Special Case

In Homogeneous permutations, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous $n$-dimensional permutation structures (homogeneous structures with $n$ linear orders) for all finite $n$. We prove here that the lattice of $\emptyset$-definable equivalence relations in such a structure can be any finite distributive lattice, providing many new imprimitive examples of homogeneous finite dimensional permutation structures. We conjecture that the distributivity of the lattice of $\emptyset$-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of $\emptyset$-definable equivalence relations is homogeneous. Finally, we conjecture a classification of the primitive examples, and confirm this in the special case where all minimal forbidden structures have order 2. 

scholarly journals k-step nilpotent Lie algebras

2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Michel Goze ◽  
Elisabeth Remm
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Deformation Process ◽  
Nilpotent Lie Algebras ◽  
Early Stages ◽  
Finite Dimensional ◽  
Contraction Process ◽  
Finite Dimensional Lie Algebras

AbstractThe classification of complex or real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example, the nilpotent Lie algebras are classified only up to dimension 7. Moreover, to recognize a given Lie algebra in the classification list is not so easy. In this work, we propose a different approach to this problem. We determine families for some fixed invariants and the classification follows by a deformation process or a contraction process. We focus on the case of 2- and 3-step nilpotent Lie algebras. We describe in both cases a deformation cohomology for this type of algebras and the algebras which are rigid with respect to this cohomology. Other

Classification of AF Flows

2003 ◽  
Vol 46 (2) ◽  
pp. 164-177 ◽  
Author(s):  
Andrew J. Dean
Keyword(s):  
Dynamical Systems ◽  
Automorphism Group ◽  
Bratteli Diagram ◽  
Finite Dimensional

AbstractAn AF flow is a one-parameter automorphism group of an AF C*-algebra A such that there exists an increasing sequence of invariant finite dimensional sub-C*-algebras whose union is dense in A. In this paper, a classification of C*-dynamical systems of this form up to equivariant isomorphism is presented. Two pictures of the actions are given, one in terms of a modified Bratteli diagram/pathspace construction, and one in terms of a modified K0 functor.

scholarly journals Equivariant Map Queer Lie Superalgebras

2016 ◽  
Vol 68 (2) ◽  
pp. 258-279 ◽  
Author(s):  
Lucas Calixto ◽  
Adriano Moura ◽  
Alistair Savage
Keyword(s):  
Finite Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Equivariant Map ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Regular Maps ◽  
A Finite Group ◽  
Special Case

AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

scholarly journals Golod–Shafarevich-Type Theorems and Potential Algebras

2018 ◽  
Vol 2019 (15) ◽  
pp. 4822-4844 ◽  
Author(s):  
Natalia Iyudu ◽  
Agata Smoktunowicz
Keyword(s):  
Linear Growth ◽  
Complete Classification ◽  
Koszul Complex ◽  
Model Program ◽  
Minimal Model Program ◽  
Infinite Dimensional ◽  
Homogeneous Potential ◽  
Finite Dimensional ◽  
The Difference

Abstract Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree $n\geqslant 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class $\mathcal {P}_{n}$ of potential algebras with homogeneous potential of degree $n+1\geqslant 4$, the minimal Hilbert series is $H_{n}=\frac {1}{1-2t+2t^{n}-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

scholarly journals Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components

1998 ◽  
Vol 40 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Zygmunt Pogorzały
Keyword(s):  
Unit Element ◽  
Global Dimension ◽  
Closed Field ◽  
Modular Representation Theory ◽  
Stable Category ◽  
Finite Global Dimension ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Stable Categories

Throughout the paper K denotes a fixed algebraically closed field. All algebras considered are finite-dimensional associative K-algebras with a unit element. Moreover, they are assumed to be basic and connected. For an algebra A we denote by mod(A) the category of all finitely generated right A-modules, and mod(A) denotes the stable category of mod(A), i.e. mod(A)/℘ where ℘ is the two-sided ideal in mod(A) of all morphisms that factorize through projective A-modules. Two algebras A and B are said to be stably equivalent if the stable categories mod(A) and mod(B) are equivalent. The study of stable equivalences of algebras has its sources in modular representation theory of finite groups. It is of importance in this theory whether two stably equivalent algebras have the same number of pairwise non-isomorphic nonprojective simple modules. Another motivation for studying stable equivalences appears in the following context. If E is a K-algebra of finite global dimension then its derived category Db(E) is equivalent to the stable category mod(Ê) of the repetitive category Ê of E [15]. Thus the problem of a classification of derived equivalent algebras leads in many cases to a classification of stably equivalent selfinjective algebras.

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