Search

Given a sequence d~ = (d1, . . . , dk) of natural numbers, we consider the Lie subalgebra h of gl(d, F), where d = d1 + · · · + dk and F is a field of characteristic 0, generated by two block upper triangular matrices D and E partitioned according to d~, and study the problem of computing the nilpotency degree m of the nilradical n of h. We obtain a complete answer when D and E belong to a certain family of matrices that arises naturally when attempting to classify the indecomposable modules of certain solvable Lie algebras. Our determination of m depends in an essential manner on the symmetry of E with respect to an outer automorphism of sl(d). The proof that m depends solely on this symmetry is long and delicate. As a direct application of our investigations on h and n we give a full classification of all uniserial modules of an extension of the free ℓ-step nilpotent Lie algebra on n generators when F is algebraically closed. Fil: Cagliero, Leandro Roberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina Fil: Levstein, Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina Fil: Szechtman, Fernando. University Of Regina; Canadá

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0 and $A\in \mathfrak{s}$ , then the semisimple and nilpotent summands of the Jordan–Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in \mathfrak{s}$ , $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$ ) such that $A=S+N$ .

Given a finite dimensional Lie algebra $\mathfrak{g}$, let $\mathfrak{z}(\mathfrak{g})$ denote the center of $\mathfrak{g}$ and let $\mu(\mathfrak{g})$ be the minimal possible dimension for a faithful representation of $\mathfrak{g}$. In this paper we obtain $\mu(\mathcal{L}_{r,2})$, where $\mathcal{L}_{r,k}$ is the free $k$-step nilpotent Lie algebra of rank $r$. In particular we prove that $\mu(\mathcal{L}_{r,2})= \left\lceil \sqrt{2r(r-1)} \right\rceil + 2$ for $r \geq 4$. It turns out that $\mu(\mathcal{L}_{r,2}) \sim\mu\big(\mathfrak{z}(\mathcal{L}_{r,2})\big) \sim 2\sqrt{\dim\mathcal{L}_{r,2}} $ (as $r\to\infty$) and we present some evidence that this could be true for $\mathcal{L}_{r,k}$ for any $k$, this is considerably lower than the known bounds for $\mu(\mathcal{L}_{r,k})$, which are (for fixed $k$) polynomial in $\dim\mathcal{L}_{r,k}$.

Given an algebraically closed field F of characteristic 0 and an F-vector space V, let L(V) = V⊕Λ2(V) denote the free 2-step nilpotent Lie algebra associated to V. In this paper, we classify all uniserial representations of the solvable Lie algebra &= ⟨x⟩⋉L(V), where x acts on V via an arbitrary invertible Jordan block. Fil: Cagliero, Leandro Roberto. Universidad Nacional de Córdoba; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina Fil: Frez, Luis Gutiérrez. Universidad Austral de Chile; Chile Fil: Szechtman, Fernando. University Of Regina; Canadá

Given a partition $$\lambda $$? of n, the Schur functor$${\mathbb {S}}_\lambda $$S? associates to any complex vector space V, a subspace $${\mathbb {S}}_\lambda (V)$$S?(V) of $$V^{\otimes n}$$V?n. Hermite's reciprocity law, in terms of the Schur functor, states that $${\mathbb {S}}_{(p)}\left( {\mathbb {S}}_{(q)}({\mathbb {C}}^2)\right) \simeq {\mathbb {S}}_{(q)}\left( {\mathbb {S}}_{(p)}({\mathbb {C}}^2)\right) . $$S(p)S(q)(C2)?S(q)S(p)(C2). We extend this identity to many other identities of the type $${\mathbb {S}}_{\lambda }\left( {\mathbb {S}}_{\delta }({\mathbb {C}}^2)\right) \simeq {\mathbb {S}}_{\mu }\left( {\mathbb {S}}_{\epsilon }({\mathbb {C}}^2)\right) $$S?S?(C2)?SμS∈(C2).

Given a finite dimensional Lie algebra $\mathfrak{g}$, let $\Gamma_\circ(\mathfrak{g})$ be the set of irreducible $\mathfrak{g}$-modules with non-vanishing cohomology. We prove that a $\mathfrak{g}$-module $V$ belongs to $\Gamma_\circ(\mathfrak{g})$ only if $V$ is contained in the exterior algebra of the solvable radical $\mathfrak{s}$ of $\mathfrak{g}$, showing in particular that $\Gamma_\circ(\mathfrak{g})$ is a finite set and we deduce that $H^*(\mathfrak{g},V)$ is an $L$-module, where $L$ is a fixed subgroup of the connected component of $\operatorname{Aut}(\mathfrak{g})$ which contains a Levi factor. We describe $\Gamma_\circ$ in some basic examples, including the Borel subalgebras, and we also determine $\Gamma_\circ(\mathfrak{s}_n)$ for an extension $\mathfrak{s}_n$ of the 2-dimensional abelian Lie algebra by the standard filiform Lie algebra $\mathfrak{f}_n$. To this end, we described the cohomology of $\mathfrak{f}_n$. We introduce the \emph{total cohomology} of a Lie algebra $\mathfrak{g}$, as $TH^*(\mathfrak{g})=\bigoplus_{V\in \Gamma_\circ(\mathfrak{g})} H^*(\mathfrak{g},V)$ and we develop further the theory of linear deformations in order to prove that the total cohomology of a solvable Lie algebra is the cohomology of its nilpotent shadow. Actually we prove that $\mathfrak{s}$ lies, in the variety of Lie algebras, in a linear subspace of dimension at least $\dim (\mathfrak{s}/\mathfrak{n})^2$, $\mathfrak{n}$ being the nilradical of $\mathfrak{s}$, that contains the nilshadow of $\mathfrak{s}$ and such that all its points have the same total cohomology., Comment: Accepted for publication in Trans. Amer. Math. Soc

We describe of all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,yin K$. When is $F[x,y]=F[alpha x+eta y]$ for some non-zero elements $alpha,etain F$? Fil: Cagliero, Leandro Roberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina Fil: Szchetman, Fernando. University of Regina; Canadá

We apply the Nash-Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space $H_{k-nil}^2(\mathfrak{g},\mathfrak{g})$, and certain subspaces of it, that provide fine information about the deformations of $\mathfrak{g}$ in the variety of $k$-step nilpotent Lie algebras. Then we focus on degenerations and rigidity in the variety of $k$-step nilpotent Lie algebras of dimension $n$ with $n\le7$ and, in particular, we obtain rigid Lie algebras and rigid curves in the variety of 3-step nilpotent Lie algebras of dimension 7. We also recover some known results and point out a possible error in a published article related to this subject., Comment: Accepted in J. of Pure and Applied Algebra. The structure of the paper has been bearly modified to follow the referee's suggestions

All Lie algebras and representations will be assumed to be finite dimensional over the complex numbers. Let V ( m ) be the irreducible sl ( 2 ) -module with highest weight m ⩾ 1 and consider the perfect Lie algebra g = sl ( 2 ) ⋉ V ( m ) . Recall that a g -module is uniserial when its submodules form a chain. In this paper we classify all uniserial g -modules. The main family of uniserial g -modules is actually constructed in greater generality for the perfect Lie algebra g = s ⋉ V ( μ ) , where s is a semisimple Lie algebra and V ( μ ) is the irreducible s -module with highest weight μ ≠ 0 . The fact that the members of this family are, but for a few exceptions of lengths 2, 3 and 4, the only uniserial sl ( 2 ) ⋉ V ( m ) -modules depends in an essential manner on the determination of certain non-trivial zeros of Racah–Wigner 6j-symbol.

With the aid of the $6j$-symbol, we classify all uniserial modules of $\mathfrak{sl}(2)\ltimes \mathfrak{h}_{n}$, where $\mathfrak{h}_{n}$ is the Heisenberg Lie algebra of dimension $2n+1$., Some references added, introduction expanded, title changed

Let g \mathfrak {g} be a finite dimensional Lie algebra over a field k k of characteristic zero. An element x x of g \mathfrak {g} is said to have an abstract Jordan-Chevalley decomposition if there exist unique s , n ∈ g s,n\in \mathfrak {g} such that x = s + n x=s+n , [ s , n ] = 0 [s,n]=0 and given any finite dimensional representation π : g → g l ( V ) \pi :\mathfrak {g}\to \mathfrak {gl}(V) the Jordan-Chevalley decomposition of π ( x ) \pi (x) in g l ( V ) \mathfrak {gl}(V) is π ( x ) = π ( s ) + π ( n ) \pi (x)=\pi (s)+\pi (n) . In this paper we prove that x ∈ g x\in \mathfrak {g} has an abstract Jordan-Chevalley decomposition if and only if x ∈ [ g , g ] x\in [\mathfrak {g},\mathfrak {g}] , in which case its semisimple and nilpotent parts are also in [ g , g ] [\mathfrak {g},\mathfrak {g}] and are explicitly determined. We derive two immediate consequences: (1) every element of g \mathfrak {g} has an abstract Jordan-Chevalley decomposition if and only if g \mathfrak {g} is perfect; (2) if g \mathfrak {g} is a Lie subalgebra of g l ( n , k ) \mathfrak {gl}(n,k) , then [ g , g ] [\mathfrak {g},\mathfrak {g}] contains the semisimple and nilpotent parts of all its elements. The last result was first proved by Bourbaki using different methods. Our proof uses only elementary linear algebra and basic results on the representation theory of Lie algebras, such as the Invariance Lemma and Lie’s Theorem, in addition to the fundamental theorems of Ado and Levi.

Let G o be a semisimple Lie group and let K o denote a maximal compact subgroup of G o . Let U ( g ) be the complex universal enveloping algebra of G o and let U ( g ) K denote the centralizer of K o in U ( g ) . Also let P : U ( g ) → U ( k ) ⊗ U ( a ) be the projection map corresponding to the direct sum U ( g ) = ( U ( k ) ⊗ U ( a ) ) ⊕ U ( g ) n associated to an Iwasawa decomposition of G o adapted to K o . In this paper we give a characterization of the image of U ( g ) K under the injective antihomomorphism P : U ( g ) K → U ( k ) ⊗ U ( a ) , considered by Lepowsky in [J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973) 1–44], when G o = Sp ( n , 1 ) .

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures.The main tools used to obtain this list are Kostant's description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained for the g-hwv's of H2(n) for a finite dimensional real symplectic nilpotent Lie algebra n with a reductive Lie subalgebra of derivations g acting on it. Fil: Cagliero, Leandro Roberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina Fil: del Barco, Viviana Jorgelina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina

We give a lower bound and an upper bound for the dimension of the homology of 2-step nilpotent Lie algebras with adjoint coefficients. We conjecture, that the upper bound and the actual dimension are asymptotically equivalent.

In this note we give an explicit closed formula for the weight multiplicities of any complex finite dimensional irreducible representation of the simple Lie group Open image in new window

Given a parabolic subalgebra g1×n of a semisimple Lie algebra, Kostant (Ann. Math. 1963) and Griffiths (Acta Math. 1963) independently computed the g1 invariants in the cohomology group of n with exterior adjoint coefficients. By a theorem of Bott (Ann. Math. 1957), this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms. In this paper we determine explicitly the full module structure, over the symplectic group, of the cohomology group of the Heisenberg Lie algebra with exterior adjoint coefficients. This is the cohomology of the cotangent bundle of the Heisenberg group.

Let G○=K○A○N○ be an Iwasawa decomposition of a connected, noncompact real semisimple Lie group with finite center and let M○ be the centralizer of A○ in K○. B. Kostant proved that for every irreducible M-spherical K-module V there exists a unique d (the Kostant degree of V) such that V can be realized as a submodule of the space of all -harmonic homogeneous polynomials of degree d on . Here is a Cartan decomposition of the complexification of the Lie algebra of G○.In this paper we give an algorithm to obtain a highest weight vector from any M-invariant vector in an irreducible M-spherical K-module. This algorithm allows us to compute a sharp bound for the Kostant degree d(v) of any M-invariant vector v in a locally finite M-spherical K-module V. The method computes d(v) effectively for any V if G○ is locally isomorphic to SO(n,1) and for if G○ is locally isomorphic to SU(n,1).

In this paper we determine the homology of the free 2-step nilpotent complex Lie algebra, with adjoint coefficients, as a module over the general linear group. This module is not multiplicity free. We give an explicit formula for the multiplicities and we compute the total dimension.

Let $F$ be an algebraically closed field and consider the Lie algebra ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$, where $\mathrm{ad}\, x$ acts diagonalizably on the abelian Lie algebra ${\mathfrak a}$. Refer to a ${\mathfrak g}$-module as admissible if $[{\mathfrak g},{\mathfrak g}]$ acts via nilpotent operators on it, which is automatic if $\mathrm{char}(F)=0$. In this paper we classify all indecomposable ${\mathfrak g}$-modules $U$ which are admissible as well as uniserial, in the sense that $U$ has a unique composition series.

Let ${\mathfrak g}$ be a finite dimensional Lie algebra over a field of characteristic 0, with solvable radical ${\mathfrak r}$ and nilpotent radical ${\mathfrak n}=[{\mathfrak g},{\mathfrak r}]$. Given a finite dimensional ${\mathfrak g}$-module $U$, its nilpotency series $ 0\subset U({\mathfrak n}^1)\subset\cdots\subset U({\mathfrak n}^m)=U$ is defined so that $U({\mathfrak n}^1)$ is the 0-weight space of ${\mathfrak n}$ in $U$, $U({\mathfrak n}^2)/U({\mathfrak n}^1)$ is the 0-weight space of ${\mathfrak n}$ in $U/U({\mathfrak n}^1)$, and so on. We say that $U$ is linked if each factor of its nilpotency series is a uniserial ${\mathfrak g}/{\mathfrak n}$-module, i.e., its ${\mathfrak g}/{\mathfrak n}$-submodules form a chain. Every uniserial ${\mathfrak g}$-module is linked, every linked ${\mathfrak g}$-module is indecomposable with irreducible socle, and both converse fail. In this paper we classify all linked ${\mathfrak g}$-modules when ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$ and $\mathrm{ad}\, x$ acts diagonalizably on the abelian Lie algebra ${\mathfrak a}$. Moreover, we identify and classify all uniserial ${\mathfrak g}$-module amongst them.

We describe the structure of the cohomology of the filiform Lie algebras and as a module over their (2-dimensional) torus of derivations. Our approach relies on the fact that both filiform algebras have an ideal of codimension 1 for which the structure of its cohomology under the action of the Levi factor of the algebra of derivations of is known. publishedVersion 2099-01 Fil: Cagliero, Leandro Roberto. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Fil: Cagliero, Leandro Roberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Cagliero, Leandro Roberto. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Tirao, Paulo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Fil: Tirao, Paulo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Tirao, Paulo. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Matemática pura

In this paper we present a lower bound for the minimal dimension $\mu(\mathfrak{n})$ of a faithful representation of a finite dimensional $p$-step nilpotent Lie algebra $\mathfrak{n}$ over a field of characteristic zero. Our bound is given as the minimum of a quadratically constrained linear optimization problem, it works for arbitrary $p$ and takes into account a given filtration of $\mathfrak{n}$. We present some estimates of this minimum which leads to a very explicit lower bound for $\mu(\mathfrak{n})$ that involves the dimensions of $\mathfrak{n}$ and its center. This bound allows us to obtain $\mu(\mathfrak{n})$ for some families of nilpotent Lie algebras.

For a two-parameter family of lower triangular matrices with entries involving Jacobi polynomials an explicit inverse is given, with entries involving a sum of two Jacobi polynomials. The formula simplifies in the Gegenbauer case and then one choice of the parameter solves an open problem in a recent paper by Koelink, van Pruijssen & Roman. The two-parameter family is closely related to two two-parameter groups of lower triangular matrices, of which we also give the explicit generators. Another family of pairs of mutually inverse lower triangular matrices with entries involving Jacobi polynomials, unrelated to the family just mentioned, was given by J. Koekoek & R. Koekoek (1999). We show that this last family is a limit case of a pair of connection relations between Askey-Wilson polynomials having one of their four parameter in common., v5: minor corrections; remark added; to appear in J. Approx. Theory

If $\mathfrak{n}$ is a $\mathbb{Z}^d_+$-graded nilpotent finite dimensional Lie algebra over a field of characteristic zero, it is well known that $\dim H^{\ast }(\mathfrak{n})\geq L(p) $ where $p$ is the polynomial associated to the grading and $L(p)$ is the sum of the absolute values of the coefficients of $p$. From this result Deninger and Singhof derived the Toral Rank Conjecture (TRC) for 2-step nilpotent Lie algebras. An algebraic version of the TRC states that $\dim H^{\ast }(\mathfrak{n})\geq 2^{\dim (\mathfrak{z)}} $ for any finite dimensional Lie algebra $\mathfrak{n}$ with center $\mathfrak{z}$. The TRC is more that 25 years old and remains open even for $\mathbb{Z}^d_+$-graded 3-step nilpotent Lie algebras. Investigating to what extent the above bound for $\dim H^{\ast }(\mathfrak{n})$ could help to prove the TRC in this case, we considered the following two questions regarding a nilpotent Lie algebra $\mathfrak{n}$ with center $\mathfrak{z}$: (A) If $\mathfrak{n}$ admits a $\mathbb{Z}_+^d$-grading $\mathfrak{n}=\bigoplus_{��\in\mathbb{Z}_+^d} \mathfrak{n}_��$, such that its associated polynomial $p$ satisfies $L(p)>2^{\dim\mathfrak{z}}$, does it admit a grading $\mathfrak{n}=\mathfrak{n}'_{1}\oplus \mathfrak{n}'_{2}\oplus \dots\oplus \mathfrak{n}'_{k}$ such that its associated polynomial $p'$ satisfies $L(p')>2^{\dim\mathfrak{z}}$? (B) If $\mathfrak{n}$ is $r$-step nilpotent admitting a grading $\mathfrak{n}=\mathfrak{n}_{1}\oplus \mathfrak{n}_{2}\oplus \dots\oplus \mathfrak{n}_{k}$ such that its associated polynomial $p$ satisfies $L(p)>2^{\dim\mathfrak{z}}$, does it admit a grading $\mathfrak{n}=\mathfrak{n}'_{1}\oplus \mathfrak{n}'_{2}\oplus \dots\oplus \mathfrak{n}'_{r}$ such that its associated polynomial $p'$ satisfies $L(p')>2^{\dim\mathfrak{z}}$? In this paper we show that the answer to (A) is yes, but the answer to (B) is no., Revised version, major modifications

Given a Lie algebra $\mathfrak{g}$ over a field of characteristic zero $k$, let $\mu(\mathfrak{g})=\min\{\dim \pi: \pi\text{is a faithful representation of}\mathfrak{g}\}$. Let $\mathfrak{h}_{m}$ be the Heisenberg Lie algebra of dimension $2m+1$ over $k$ and let $k[t]$ be the polynomial algebra in one variable. Given $m\in\mathbb{N}$ and $p\in k[t]$, let $\mathfrak{h}_{m,p}=\mathfrak{h}_m\otimes k[t]/(p)$ be the current Lie algebra associated to $\mathfrak{h}_m$ and $k[t]/(p)$, where $(p)$ is the principal ideal in $k[t]$ generated by $p$. In this paper we prove that $ mu(\mathfrak{h}_{m,p}) = m \deg p + \left \lceil 2\sqrt{\deg p} \right\rceil$., Comment: 14 pages

A main contribution of this paper is the explicit construction of comparison morphisms between the standard bar resolution and Bardzell's minimal resolution for truncated quiver algebras (TQA's). As a direct application we describe explicitely the Yoneda product and derive several results on the structure of the cohomology ring of TQA's. For instance, we show that the product of odd degree cohomology classes is always zero. We prove that TQA's associated with quivers with no cycles or with neither sinks nor sources have trivial cohomology rings. On the other side we exhibit a fundamental example of a TQA with non trivial cohomology ring. Finaly, for truncated polyniomial algebras in one variable, we construct explicit cohomology classes in the bar resolution and give a full description of their cohomology ring., Comment: 32 pages, Final Version

The finite dimensional complex truncated tensor algebras have a natural module structure over the complex general linear group. This structure is inherited by the Hochschild homology of these algebras. In this paper we determine this module structure by combining techniques from homological algebra and representation theory, such as the Schur duality theorem.

We define a quiver to be rigid if all the associated truncated quiver algebras are rigid. The rigidity of quivers is then determined by the combinatorics of the set of pairs of parallel paths of the underlying quiver as follows from Cibils' criteria for the rigidity of truncated quiver algebras. In this paper we characterize rigid quivers Δ and relate this characterization with the condensed quiver and the quiver of beads of Δ, two much simpler quivers associated to Δ. The first one is a well-known object and the second one is introduced by us to this end.

Tilapia nilotica is commercially very important throughout the Ethiopian region including the major rivers in West Africa, the Chad basin, the Nile and its associated lakes. The Tilapia fishery of Lake Rudolf is at present small, but potentially important, particularly on the eastern shores of the lake where fishing intensity is low. Preliminary results from observations on the feeding of Tilapia nilotica in Lake Rudolf are presented. The fish exhibit a regular diurnal feeding rhythm, commencing between 05.00 hours and 08.00 hours and ceasing between 14.00 hours and 18.00 hours. The largest fish appear to feed longer. Quantitative estimates of the daily food intake indicate less material to be ingested than by populations in other lakes. The lysis of algae, intestinal pH and food material are also investigated.