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Register a new userSpectral invariants for finite dimensional Lie algebras
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For a Lie algebra ${mathcal L}$ with basis ${x_1,x_2,cdots,x_n}$, its associated characteristic polynomial $Q_{{mathcal L}}(z)$ is the determinant of the linear pencil $z_0I+z_1text{ad} x_1+cdots +z_ntext{ad} x_n.$ This paper shows that $Q_{mathcal L}$ is invariant under the automorphism group $text{Aut}({mathcal L}).$ The zero variety and factorization of $Q_{mathcal L}$ reflect the structure of ${mathcal L}$. In the case ${mathcal L}$ is solvable $Q_{mathcal L}$ is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincar{e} polynomial for solvable Lie algebras. Application is given to $1$-dimensional extensions of nilpotent Lie algebras.
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The deformed current Lie algebra was introduced by the author to study the representation theory of cyclotomic q-Schur algebras at q=1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.
In the article at hand, we sketch how, by utilizing nilpotency to its fullest extent (Engel, Super Engel) while using methods from the theory of universal enveloping algebras, a complete description of the indecomposable representations may be reached. In practice, the combinatorics is still formidable, though. It turns out that the method applies to both a class of ordinary Lie algebras and to a similar class of Lie superalgebras. Besides some examples, due to the level of complexity we will only describe a few precise results. One of these is a complete classification of which ideals can occur in the enveloping algebra of the translation subgroup of the Poincare group. Equivalently, this determines all indecomposable representations with a single, 1-dimensional source. Another result is the construction of an infinite-dimensional family of inequivalent representations already in dimension 12. This is much lower than the 24-dimensional representations which were thought to be the lowest possible. The complexity increases considerably, though yet in a manageable fashion, in the supersymmetric setting. Besides a few examples, only a subclass of ideals of the enveloping algebra of the super Poincare algebra will be determined in the present article.
Let $k$ be a field and let $Lambda$ be a finite dimensional $k$-algebra. We prove that every bounded complex $V^bullet$ of finitely generated $Lambda$-modules has a well-defined versal deformation ring $R(Lambda,V^bullet)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. We also prove that nice two-sided tilting complexes between $Lambda$ and another finite dimensional $k$-algebra $Gamma$ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra.
A host algebra of a (possibly infinite dimensional) Lie group $G$ is a $C^*$-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations $pi colon G to U(cH)$. In this paper we present a new approach to host algebras for infinite dimensional Lie groups which is based on smoothing operators, i.e., operators whose range is contained in the space $cH^infty$ of smooth vectors. Our first major result is a characterization of smoothing operators $A$ that in particular implies smoothness of the maps $pi^A colon G to B(cH), g mapsto pi(g)A$. The concept of a smoothing operator is particularly powerful for representations $(pi,cH)$ which are semibounded, i.e., there exists an element $x_0 ing$ for which all operators $iddpi(x)$, $x in g$, from the derived representation are uniformly bounded from above in some neighborhood of $x_0$. Our second main result asserts that this implies that $cH^infty$ coincides with the space of smooth vectors for the one-parameter group $pi_{x_0}(t) = pi(exp tx_0)$. We then show that natural types of smoothing operators can be used to obtain host algebras and that, for every metrizable Lie group, the class of semibounded representations can be covered completely by host algebras. In particular, it permits direct integral decompositions.
In this thesis new objects to the existing set of invariants of Lie algebras are added. These invariant characteristics are capable of describing the nilpotent parametric continuum of Lie algebras. The properties of these invariants, in view of possible alternative classifications of Lie algebras, are formulated and their behaviour on known lower--dimensional Lie algebras investigated. It is demonstrated that these invariants, in view of their application on graded contractions of sl(3,C), are also effective in higher dimensions. A necessary contraction criterion involving these invariants is derived and applied to lower--dimensional cases. Possible application of these invariant characteristics to Jordan algebras is investigated.
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