scholarly journals GENERIC IRREDUCIBLE REPRESENTATIONS OF FINITE-DIMENSIONAL LIE SUPERALGEBRAS

1994 ◽  
Vol 05 (03) ◽  
pp. 389-419 ◽  
Author(s):  
IVAN PENKOV ◽  
VERA SERGANOVA
Keyword(s):  
Irreducible Module ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Irreducible Representations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Necessary And Sufficient

A theory of highest weight modules over an arbitrary finite-dimensional Lie superalgebra is constructed. A necessary and sufficient condition for the finite-dimensionality of such modules is proved. Generic finite-dimensional irreducible representations are defined and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic finite-dimensional irreducible module over any finite-dimensional Lie superalgebra. The conjecture is proved for several classes of Lie superalgebras, in particular for all solvable ones, for all simple ones, and for certain semi-simple ones.

scholarly journals On the forces that cable webs under tension can support and how to design cable webs to channel stresses

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180781 ◽  
Author(s):  
Guy Bouchitté ◽  
Ornella Mattei ◽  
Graeme W. Milton ◽  
Pierre Seppecher
Keyword(s):  
Linear Programming ◽  
Convex Hull ◽  
Structural Engineering ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In many applications of structural engineering, the following question arises: given a set of forces f 1 ,  f 2 , …,  f N applied at prescribed points x 1 ,  x 2 , …,  x N , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x 1 ,  x 2 , …,  x N in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f 1 ,  f 2 , …,  f N applied at points strictly within the convex hull of x 1 ,  x 2 , …,  x N . In three dimensions, we show that, by slightly perturbing f 1 ,  f 2 , …,  f N , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for channelling stress in desired ways.

scholarly journals Four-dimensional vector multiplets in arbitrary signature (I)

2020 ◽  
Vol 17 (10) ◽  
pp. 2050150 ◽  
Author(s):  
V. Cortés ◽  
L. Gall ◽  
T. Mohaupt
Keyword(s):  
Vector Space ◽  
Classification Problem ◽  
Lie Superalgebras ◽  
Symmetry Groups ◽  
Sufficient Condition ◽  
Dimensional Vector ◽  
Necessary And Sufficient Condition ◽  
Arbitrary Signature ◽  
Necessary And Sufficient ◽  
R Symmetry

We derive a necessary and sufficient condition for Poincaré Lie superalgebras in any dimension and signature to be isomorphic. This reduces the classification problem, up to certain discrete operations, to classifying the orbits of the Schur group on the vector space of superbrackets. We then classify four-dimensional [Formula: see text] supersymmetry algebras, which are found to be unique in Euclidean and in neutral signature, while in Lorentz signature there exist two algebras with R-symmetry groups [Formula: see text] and [Formula: see text], respectively.

scholarly journals CONNECTIVITY PROPERTIES OF MCKAY QUIVERS

2020 ◽  
pp. 1-13
Author(s):  
HAZEL BROWNE
Keyword(s):  
Finite Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Self Duality ◽  
Finite Dimensional ◽  
Representations Of Finite Groups ◽  
Necessary And Sufficient ◽  
Mckay Quiver

Abstract We present several results on the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver.

scholarly journals Polynomial identities for simple Lie superalgebras

1987 ◽  
Vol 28 (3) ◽  
pp. 310-327 ◽  
Author(s):  
M. D. Gould
Keyword(s):  
Tensor Product ◽  
Lie Superalgebra ◽  
Weight Vector ◽  
Lie Superalgebras ◽  
Irreducible Representations ◽  
Polynomial Identities ◽  
Minimal Weight ◽  
Finite Dimensional ◽  
Basic Classical Lie Superalgebra

AbstractPolynomial identities for the generators of a simple basic classical Lie superalgebra are derived in arbitrary representations generated by a maximal (or minimal) weight vector. The infinitesimal characters occurring in the tensor product of two finite dimensional irreducible representations are also determined.

On generalized inverses of matrices

1966 ◽  
Vol 62 (4) ◽  
pp. 673-677 ◽  
Author(s):  
M. H. Pearl
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Maximal Rank ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Finite Dimensional ◽  
The Matrix ◽  
Generalized Inverses Of Matrices ◽  
Necessary And Sufficient

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.

scholarly journals Weakly externally hyperconvex subsets and hyperconvex gluings

2016 ◽  
Vol 09 (03) ◽  
pp. 379-407
Author(s):  
Benjamin Miesch ◽  
Maël Pavón
Keyword(s):  
Metric Spaces ◽  
Maximum Norm ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Spaces ◽  
Full Characterization ◽  
Finite Dimensional ◽  
Hyperconvex Space ◽  
Necessary And Sufficient

We give a necessary and sufficient condition under which gluings of hyperconvex metric spaces along weakly externally hyperconvex subsets are hyperconvex. This leads to a full characterization of hyperconvex gluings of two isometric copies of the same hyperconvex space. Furthermore, we investigate the case of gluings of finite dimensional hyperconvex linear spaces along linear subspaces. For this purpose, we characterize the weakly externally hyperconvex subsets of [Formula: see text] endowed with the maximum norm.

LIMITS OF HOMOMORPHISMS WITH FINITE-DIMENSIONAL RANGE

2005 ◽  
Vol 16 (07) ◽  
pp. 807-821 ◽  
Author(s):  
SHANWEN HU ◽  
HUAXIN LIN ◽  
YIFENG XUE
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
C Algebra ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Unital Simple ◽  
Dimensional Range

Let X be a compact metric space and A be a unital simple C*-algebra with TR (A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.

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