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Donald W. Barnes

Publications and source records attributed to Donald W. Barnes.

At least 19 recordsLinked to original sources
arXiv2019

Faithful irreducible representations of modular Lie algebras

Let L be a finite-dimensional Lie algebra over a field of non-zero characteristic. By a theorem of Jacobson, L has a finite-dimensional faithful module which is completely reducible. We show that if the field is not algebraically closed, then L has an irreducible such module. We also give a necessary and sufficient condition for a finite-dimensional Lie algebra over an algebraically closed field of non-zero characteristic to have a faithful irreducible module.

math.RT↗
arXiv2016

Induced modules for modular Lie algebras

Let $L$ be a finite-dimensional Lie algebra over a field of non-zero characteristic and let $S$ be a subalgebra. Suppose that $X$ is a finite set of finite-dimensional $L$-modules. Let $D$ be the category of all finite-dimensional $S$-modules. Then there exists a category $C$ of finite-dimensional $L$-modules containing the modules in $X$ such that the restriction functor Res$:C \to D$ has a left adjoint Ind$:D \to C$.

math.RA↗
arXiv2016

On $\mathfrak F$-hypercentral modules and character clusters

Let $\mathfrak F$ be a saturated formation of soluble Lie algebras over a field $F$ of characteristic $p > 0$ and let ${\mathbb F}_p$ denote the field of $p$ elements. Let $(L,[p])$ be a restricted Lie algebra over $F$ with $z^{$\scriptstyle [p]$}=0$ for all $z$ in the centre of $L$. Let $S \in \mathfrak F$, $S\ne 0$ be a subnormal subalgebra of $L$. Let $V, W$ be $L$-modules. Suppose that the character cluster of $W$ is contained in the set of ${\mathbb F}_p$-linear combinations of the characters in the character cluster of $V$. Suppose that $V$, regarded as $S$-module, is $\mathfrak F$-hypercentral. Then $W$, regarded as $S$-module, is also $\mathfrak F$-hypercentral.

math.RA↗
arXiv2016

Faithful completely reducible representations of modular Lie algebras

The Ado-Iwasawa Theorem asserts that a finite-dimensional Lie algebra $L$ over a field $F$ has a finite-dimensional faithful module $V$. There are several extensions asserting the existence of such a module with various additional properties. In particular, Jacobson has proved that if the field has characteristic $p>0$, then there exists a completely reducible such module $V$. I strengthen Jacobson's Theorem, proving that if $L$ has dimension $n$ over the field $F$ of characteristic $p>0$, then $L$ has a faithful completely reducible module $V$ with $\dim(V) \le p^{n^2-1}$.

math.RA↗
arXiv2015

Blocks of representations of Lie algebras

In the theory of finite groups, the irreducible representations of G over a field F are classified into blocks based on a direct decompositions of the group algebra FG. This gives a natural decomposition of FG-modules into direct summands, each summand having all its composition factors belonging to a single block. This block decomposition is the finest natural decomposition of the FG-modules. In this paper, a classification of the irreducible representations of a finite dimensional Lie algebra L into blocks is defined, giving the finest natural direct decomposition of L-modules. This classification is investigated for supersoluble algebras.

math.RA↗
arXiv2013

Character clusters for Lie algebra modules over a field of non-zero characteristic

For a Lie algebra L over an algebraically closed field of non-zero characteristic, every finite-dimensional L-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Clusters are used to generalise the construction of induced modules.

math.RA↗
arXiv2012

Strong containment of saturated formations of soluble Lie algebras

It is shown that, if H,K are saturated formations of soluble Lie algebras over a field of non-zero characteristic and H strongly contains K non-trivially, then H coincides with the formation generated by the L/N(L) for L in H, and that H is not locally defined.

math.RA↗
arXiv2011

On locally defined formations of soluble Lie and Leibniz algebras

It is well-known that all saturated formations of finite soluble groups are locally defined and, except for the trivial formation, have many different local definitions. I show that for Lie and Leibniz algebras over a field of characteristic 0, the formations of all nilpotent algebras and of all soluble algebras are the only locally defined formations and that the latter has many local definitions. Over a field of non-zero characteristic, a saturated formation of soluble Lie algebras has at most one local definition but a locally defined saturated formation of soluble Leibniz algebras other than that of nilpotent algebras has more than one local definition.

math.RA↗
arXiv2011

Chief factors covered by projectors of soluble Leibniz algebras

Let F be a saturated formation of soluble Leibniz algebras. Let K be an F-projector and A/B a chief factor of the soluble Leibniz algebra L. It is well-known that if A/B is F-central, then K covers A/B. I prove the converse: if K covers A/B, then A/B is F-central.

math.RA↗
arXiv2011

On Levi's Theorem for Leibniz algebras

A Lie algebra over a field of characteristic 0 splits over its soluble radical and all complements are conjugate. I show that the splitting theorem extends to Leibniz algebras but that the conjugacy theorem does not.

math.RA↗
arXiv2011

Lattices of subalgebras of Leibniz algebras

I describe the lattice of subalgebras of a one-generator Leibniz algebra. Using this, I show that, apart from one special case, a lattice isomorphism between Leibniz algebras L, L' maps the Leibniz kernel of L to that of L'.

math.RA↗
arXiv2011

Schunck classes of soluble Leibniz algebras

I set out the theory of Schunck classes and projectors for soluble Leibniz algebras, parallel to that for Lie algebras. Primitive Leibniz algebras come in pairs, one (Lie) symmetric, the other antisymmetric. A Schunck formation containing one member of a pair also contains the other. Projectors for a Schunck formation are intravariant.

math.RA↗
arXiv2008

Subnormal subalgebras of Leibniz algebras

Zassenhaus has proved that if U is a subnormal subalgebra of a finite-dimensional Lie algebra L and V is a finite-dimensional irreducible L-module, then all U-module composition factors of V are isomorphic. Schenkman has proved that if U is a subnormal subalgebra of a finite-dimensional Lie algebra L, then the nilpotent residual of U is an ideal of L. These useful results generalise to Leibniz algebras.

math.RA↗