Triangular representations of linear algebras

1953 ◽  
Vol 49 (4) ◽  
pp. 595-600 ◽  
Author(s):  
M. P. Drazin
Keyword(s):  
Commutative Algebra ◽  
Similarity Transformation ◽  
Closed Field ◽  
Triangular Form ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Common Eigenvector ◽  
General Systems ◽  
The Matrix ◽  
Matrix Case

It is well known that the elements of any given commutative algebra (and hence of any commutative set) of n × n matrices, over an algebraically closed field K, have a common eigenvector over K; indeed, the elements of such an algebra can be simultaneously reduced to triangular form (by a suitable similarity transformation). McCoy (5) has shown that a triangular reduction is always possible even for matrix algebras satisfying a condition substantially weaker than commutativity. Our aim in this note is to extend these results to more general systems (our arguments being, incidentally, simpler than some used for the matrix case even by writers subsequent to McCoy).

Abelian Gradings on Upper Block Triangular Matrices

2012 ◽  
Vol 55 (1) ◽  
pp. 208-213 ◽  
Author(s):  
Angela Valenti ◽  
Mikhail Zaicev
Keyword(s):  
Abelian Group ◽  
Characteristic Zero ◽  
Finite Abelian Group ◽  
Triangular Matrix ◽  
Closed Field ◽  
Triangular Matrices ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Block Triangular Matrix

AbstractLet G be an arbitrary finite abelian group. We describe all possible G-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.

Graded Involutions on Upper-triangular Matrix Algebras

2009 ◽  
Vol 16 (01) ◽  
pp. 103-108 ◽  
Author(s):  
A. Valenti ◽  
M. V. Zaicev
Keyword(s):  
Abelian Group ◽  
Characteristic Zero ◽  
Finite Abelian Group ◽  
Triangular Matrix ◽  
Closed Field ◽  
Triangular Matrices ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Upper Triangular Matrices ◽  
Upper Triangular Matrix

Let UTn be the algebra of n × n upper-triangular matrices over an algebraically closed field of characteristic zero. We describe all G-gradings on UTn by a finite abelian group G commuting with an involution (involution gradings).

scholarly journals The Matrix Equation XA- AX=Xαg(X) over Fields or Rings

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Gerald Bourgeois
Keyword(s):  
Commutative Ring ◽  
Matrix Equation ◽  
Zero Divisor ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Reduced Ring ◽  
The Matrix ◽  
Generic Matrix

Let n,α∈N≥2 and let K be an algebraically closed field with characteristic 0 or greater than n. We show that if f∈K[X] and A,B∈Mn(K) satisfy [A,B]=f(A), then A,B are simultaneously triangularizable. Let R be a reduced ring such that n! is not a zero divisor and let A be a generic matrix over R; we show that X=0 is the sole solution of AX-XA=Xα. Let R be a commutative ring with unity; let A be similar to diag(λ1In1,…,λrInr) such that, for every i≠j,λi-λj is not a zero divisor. If X is a nilpotent solution of XA-AX=Xαg(X) where g∈R[X], then AX=XA.

scholarly journals Pairs of matrices with a non-zero commutator

1955 ◽  
Vol 51 (4) ◽  
pp. 551-553 ◽  
Author(s):  
L. S. Goddard ◽  
H. Schneider
Keyword(s):  
Characteristic Root ◽  
Column Vector ◽  
Closed Field ◽  
Complex Matrix ◽  
Algebraically Closed Field ◽  
Characteristic Roots ◽  
Linearly Independent ◽  
Special Cases ◽  
The Matrix ◽  
Square Complex

1. This note takes its origin in a remark by Brauer (1) and Perfect (5): Let A be a square complex matrix of order n whose characteristic roots are α1,…, αn. If X1 is a characteristic column vector with associated root α and k is any row vector, then the characteristic roots of A + X1 k are α1 + KX1, α2, …, αn. Recently, Goddard (2) extended this result as follows: If x1; …, xr are linearly independent characteristic column vectors associated with the characteristic roots α1, …, αr of the matrix A, whose elements lie in any algebraically closed field, then any characteristic root of Λ + KX is also a characteristic root of A + XK, where K is an arbitrary r × n matrix, X = (x1, …, xr) and Λ = diag (α1, …, αr). We shall prove some theorems of which these and other well-known results are special cases.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero 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. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

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