scholarly journals BIDIAGONAL PAIRS, THE LIE ALGEBRA 𝔰𝔩2, AND THE QUANTUM GROUP Uq(𝔰𝔩2)

2013 ◽  
Vol 12 (05) ◽  
pp. 1250207 ◽  
Author(s):  
DARREN FUNK-NEUBAUER
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Dimensional Representation ◽  
Linear Transformations ◽  
Finite Dimensional Representation ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The Relationship

We introduce a linear algebraic object called a bidiagonal pair. Roughly speaking, a bidiagonal pair is a pair of diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other. We associate to each bidiagonal pair a sequence of scalars called a parameter array. Using this concept of a parameter array we present a classification of bidiagonal pairs up to isomorphism. The statement of this classification does not explicitly mention the Lie algebra 𝔰𝔩2 or the quantum group Uq(𝔰𝔩2). However, its proof makes use of the finite-dimensional representation theory of 𝔰𝔩2 and Uq(𝔰𝔩2). In addition to the classification we make explicit the relationship between bidiagonal pairs and modules for 𝔰𝔩2 and Uq(𝔰𝔩2).

scholarly journals FAITHFUL REPRESENTATIONS OF MINIMAL DIMENSION OF CURRENT HEISENBERG LIE ALGEBRAS

2009 ◽  
Vol 20 (11) ◽  
pp. 1347-1362 ◽  
Author(s):  
LEANDRO CAGLIERO ◽  
NADINA ROJAS
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Abelian Subalgebras ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Minimal Dimension

Given a Lie algebra 𝔤 over a field of characteristic zero k, let μ(𝔤) = min{dim π : π is a faithful representation of 𝔤}. Let 𝔥m be the Heisenberg Lie algebra of dimension 2m + 1 over k and let k [t] be the polynomial algebra in one variable. Given m ∈ ℕ and p ∈ k [t], let 𝔥m, p = 𝔥m ⊗ k [t]/(p) be the current Lie algebra associated to 𝔥m and k [t]/(p), where (p) is the principal ideal in k [t] generated by p. In this paper we prove that [Formula: see text]. We also prove a result that gives information about the structure of a commuting family of operators on a finite dimensional vector space. From it is derived the well-known theorem of Schur on maximal abelian subalgebras of 𝔤𝔩(n, k ).

scholarly journals The ideal structure of idempotent-generated transformation semigroups

1985 ◽  
Vol 28 (3) ◽  
pp. 319-331 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Boolean Ring ◽  
Ideal Structure ◽  
Transformation Semigroups ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The Ideal

Let X be a set and the semigroup (under composition) of all total transformations from X into itself. In ([6], Theorem 3) Howie characterised those elements of that can be written as a product of idempotents in different from the identity. We gather from review articles that his work was later extended by Evseev and Podran [3, 4] (and independently for finite X by Sullivan [15]) to the semigroup of all partial transformations of X into itself. Howie's result was generalized in a different direction by Kim [8], and it has also been considered in both a topological and a totally ordered setting (see [11] and [14] for brief summaries of this latter work). In addition, Magill [10] investigated the corresponding idea for endomorphisms of a Boolean ring, while J. A. Erdos [2] resolved the analogous problem for linear transformations of a finite–dimensional vector space.

Linear Transformations on Symmetric Spaces II

1993 ◽  
Vol 45 (2) ◽  
pp. 357-368 ◽  
Author(s):  
Ming–Huat Lim
Keyword(s):  
Vector Space ◽  
Product Space ◽  
Symmetric Spaces ◽  
Real Field ◽  
Dimensional Vector ◽  
Infinite Field ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

AbstractLet U be a finite dimensional vector space over an infinite field F. Let U(r) denote the r–th symmetric product space over U. Let T: U(r) → U(s) be a linear transformation which sends nonzero decomposable elements to nonzero decomposable elements. Let dim U ≥ s + 1. Then we obtain the structure of T for the following cases: (I) F is algebraically closed, (II) F is the real field, and (III) T is injective.

Products of idempotent linear transformations

1985 ◽  
Vol 100 (1-2) ◽  
pp. 123-138 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bounded Operators ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

SynopsisIn 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

Invariant Polynomials of Weyl Groups and Applications to the Centres of Universal Enveloping Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 583-592 ◽  
Author(s):  
C. Y. Lee
Keyword(s):  
Lie Algebra ◽  
Weyl Groups ◽  
Killing Form ◽  
Dimensional Representation ◽  
Universal Enveloping Algebras ◽  
Semisimple Lie Algebra ◽  
Finite Dimensional Representation ◽  
Polynomial Invariants ◽  
Finite Dimensional ◽  
Complete Set

An element in the centre of the universal enveloping algebra of a semisimple Lie algebra was first constructed by Casimir by means of the Killing form. By Schur's lemma, in an irreducible finite-dimensional representation elements in the centre are represented by diagonal matrices of all whose eigenvalues are equal. In section 2 of this paper, we show the existence of a complete set of generators whose eigenvalues in an irreducible representation are closely related to polynomial invariants of the Weyl group W of the Lie algebra (Theorem 1).

CONSTRUCTING FULL BLOCK TRIANGULAR REPRESENTATIONS OF ALGEBRAS

2007 ◽  
Vol 06 (02) ◽  
pp. 259-265
Author(s):  
DANIEL BIRMAJER
Keyword(s):  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Direct Sum ◽  
Indecomposable Representations ◽  
Finite Dimensional ◽  
Triangular Representation ◽  
Representations Of Algebras ◽  
Finitely Presented ◽  
Complicated Task

Every finite dimensional representation of an algebra is equivalent to a finite direct sum of indecomposable representations. Hence, the classification of indecomposable representations of algebras is a relevant (and usually complicated) task. In this note we study the existence of full block triangular representations, an interesting example of indecomposable representations, from a computational perspective. We describe an algorithm for determining whether or not an associative finitely presented k-algebra R has a full block triangular representation over [Formula: see text].

scholarly journals Principal Irreducible Lie-Algebra Modules

1978 ◽  
Vol 21 (4) ◽  
pp. 483-489 ◽  
Author(s):  
Frank J. Servedio
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Let V be a finite dimensional vector space over k, a field of characteristic 0, L be an algebraic Lie-subalgebra of Endk(V), with the latter a Lie algebra in the canonical way, and let V be an L-module in the canonical way. For X ∈ V, let LX = {AX | A ∈ L{. Call V a principal L-module if ∃ X ∈ V such that LX= V; X will be called a principal generator of the L-module V.

scholarly journals Invariants of Certain Groups I

1971 ◽  
Vol 41 ◽  
pp. 69-73 ◽  
Author(s):  
Takehiko Miyata
Keyword(s):  
Vector Space ◽  
Symmetric Algebra ◽  
Transcendence Degree ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Rational Field ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Natural Inclusion

Let G be a group and let k be a field. A K-representation ρ of G is a homomorphism of G into the group of non-singular linear transformations of some finite-dimensional vector space V over k. Let K be the field of fractions of the symmetric algebra S(V) of V, then G acts naturally on K as k-automorphisms. There is a natural inclusion map V→K, so we view V as a k-subvector space of K. Let v1, v2, · · ·, vn be a basis for V, then K is generated by v1, v2, · · ·, vn over k as a field and these are algebraically independent over k, that is, K is a rational field over k with the transcendence degree n. All elements of K fixed by G form a subfield of K. We denote this subfield by KG.

Rank 1 preservers on the unitary Lie ring

1990 ◽  
Vol 49 (3) ◽  
pp. 399-417 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Lie Ring ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Commuting Pairs ◽  
Additive Maps

AbstractThe surjective additive maps on the Lie ring of skew-Hermitian linear transformations on a finite-dimensional vector space over a division ring which preserve the set of rank 1 elements are determined. As an application, maps preserving commuting pairs of transformations are determined.

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