Faithful Finite Dimensional Representation Research Articles

Universal enveloping of a graded Lie algebra

In this paper we construct a graded universal enveloping algebra of a G-graded Lie algebra, where G is not necessarily an abelian group. If the grading group is abelian, then it coincides with the classical construction. We prove the existence and uniqueness of the graded enveloping algebra. As consequences, we prove a graded variant of Witt's Theorem on the universal enveloping algebra of the free Lie algebra, and the graded version of Ado's Theorem, which states that every finite-dimensional Lie algebra admits a faithful finite dimensional representation. Furthermore we investigate if a Lie grading is equivalent to an abelian grading.

Bogomolov multipliers of some groups of order p 6

Let G be a finite group, V a faithful finite-dimensional representation of G over the complex field and be the corresponding invariant field. The Bogomolov multiplier of G is canonically isomorphic to the unramified cohomological group which has been used by Saltman in 1984 and Bogomolov in 1988 to provide counterexamples to the rationality problem of for finite p-groups P over In this paper, we investigate the vanishing property of where P denotes a p-group of order p 6 for

A Lax type operator for quantum finite W-algebras

For a reductive Lie algebra $$\mathfrak {g}$$ , its nilpotent element f and its faithful finite dimensional representation, we construct a Lax operator L(z) with coefficients in the quantum finite W-algebra $$W(\mathfrak {g},f)$$ . We show that for the classical linear Lie algebras $$\mathfrak {gl}_N$$ , $$\mathfrak {sl}_N$$ , $$\mathfrak {so}_N$$ and $$\mathfrak {sp}_N$$ , the operator L(z) satisfies a generalized Yangian identity. The operator L(z) is a quantum finite analogue of the operator of generalized Adler type which we recently introduced in the classical affine setup. As in the latter case, L(z) is obtained as a generalized quasideterminant.

Connected Lie groups having faithful locally bounded (not necessarily continuous) finite-dimensional representations

In the paper, it is proved that a connected Lie group admits a (possibly discontinuous) faithful finite-dimensional representation if and only if it admits a continuous faithful finite-dimensional representation, i.e., if and only if it is a linear Lie group.

Gaussian hemigroups on a locally compact group

A notion of Gaussian hemigroup is introduced and its relationship with the Gauss condition is studied. Moreover, a Levy-type martingale characterization is proved for processes with independent (not necessarily stationary) increments satisfying the Gauss condition in a compact Lie group. The characterization is given in terms of a faithful finite dimensional representation of the group and its tensor square. For the proofs noncommutative Fourier theory is applied for the convolution hemigroups associated with the increment processes.

MARTINGALE CHARACTERIZATIONS OF INCREMENT PROCESSES IN A LOCALLY COMPACT GROUP

Martingale characterizations and the related martingale problem are studied for processes with independent (not necessarily stationary) increments in an arbitrary locally compact group. In the special case of a compact Lie group, a Lévy-type characterization is given in terms of a faithful finite dimensional representation of the group and its tensor square. For the proofs noncommutative Fourier theory is applied for the convolution hemigroups associated with the increment processes.

On the finite dimensional representations of the pseudo-lie algebras

Assuming that the pseudo-Lie algebra has a faithful finite dimensional representation its structure is obtained in the case when the realization of this representation is possible by means of matrices of possibly minimal size.The results are similar to those obtained by Haag, Łopuszański and Sohnius with use of different methods.

Characters and complexity of finite semigroups

Let S be a finite semigroup and let K be an algebraically closed field of characteristic zero. Herein, we derive a formula for the congruence ≡ induced on S by the direct sum of all the irreducible representations of S over K. This congruence ≡ is proved to be the same as the congruence induced by the minimal homomorphic image of S, which is one-to-one on the subgroups of S and such that two distinct principal ideals of S, each generated by an idempotent, have distinct images. Assume ψ is a faithful finite dimensional representation of S. Let R(ψ) be the associated completely reducible representation having the same character as ψ. That is R(ψ) is the direct sum of the Jordan-Hölder factors of ψ. Then by applying the Burnside-Steinberg theorem we prove that the congruence induced by R(ψ) on S equals ≡. That is, the operator R preserves “one-to-oneness as much as possible”. We next apply these results to compute the complexity, # G ( S), of S when S is a union of groups. A linear transformation T=B(S) is defined on the character ring of S into itself. Then by a theorem of Krohn-Rhodes (which determines the complexity of S in terms of its homomorphic images), together with the previous character results, we prove that T is nilpotent and index( T)=# G ( S). Finally the character results proved here imply that, if the Fundamental Lemma on Complexity is valid, then the complexity of S is the maximum of the images of all its irreducible representations. This is known to be the case for all regular semigroups.

An addition to Ado’s theorem

For the suggestion that this nilpotency property of p might be secured I am indebted to Leonard Ross who used the characteristic 0 case of Theorem 1 in his proof of Ado's Theorem for graded Lie algebras (Thesis, Cohomology of graded lie algebras, University of California, Berkeley, 1964). In the case of characteristic 0, it is known that there exists a faithful finite-dimensional representation of L whose restriction to the maximum nilpotent ideal of L is nilpotent [1, pp. 202-203]. Hence, in order to establish Theorem 1 in the case of characteristic 0, it suffices to make the following observation: Let L be a finite-dimensional Lie algebra over a field of characteristic 0, and let M be a finite-dimensional L-module on which the maximum nilpotent ideal N of L is nilpotent. Let x be an element of L whose adjoint image a(x) is nilpotent. Then x is nilpotent on M. PROOF. Write L = S+R, where R is the radical of L and S is a semisimple subalgebra of L. Accordingly, write x=s+r, with s in S and r in R. Since a(x) is nilpotent, it is clear that the adjoint representation of S sends s onto a nilpotent derivation of S. Since S

Representations of graded Lie algebras

Introduction. Let L be a graded Lie algebra (see ?2) over a commutative ring R with unit. We shall deal with graded representations of L. In ?2 we define the universal enveloping algebra U of L and prove a PoincareBirkhoff-Witt theorem for U. In ?3 we prove that if L is a finite dimensional graded Lie algebra over a field of characteristic : 2, then L has a faithful finite dimensional representation. A key lemma for this proof is a result on (nongraded) Lie algebras which states that if L is a finite dimensional Lie algebra over a field of characteristic 0 then L has a faithful, finite dimensional representation f such that f (x) is nilpotent whenever ad x is nilpotent. This work is the first section of a doctoral thesis written under the direction of Professor Gerhard Hochschild. The author takes this opportunity to thank him for his generous advice and instruction.