Search results for "Triangular matrix"
showing 8 items of 28 documents
Abelian Gradings on Upper Block Triangular Matrices
2012
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.
Differential identities, 2 × 2 upper triangular matrices and varieties of almost polynomial growth
2019
Abstract We study the differential identities of the algebra U T 2 of 2 × 2 upper triangular matrices over a field of characteristic zero. We let the Lie algebra L = Der ( U T 2 ) of derivations of U T 2 (and its universal enveloping algebra) act on it. We study the space of multilinear differential identities in n variables as a module for the symmetric group S n and we determine the decomposition of the corresponding character into irreducibles. If V is the variety of differential algebras generated by U T 2 , we prove that unlike the other cases (ordinary identities, group graded identities) V does not have almost polynomial growth. Nevertheless we exhibit a subvariety U of V having almo…
Specht property for some varieties of Jordan algebras of almost polynomial growth
2019
Abstract Let F be a field of characteristic zero. In [25] it was proved that U J 2 , the Jordan algebra of 2 × 2 upper triangular matrices, can be endowed up to isomorphism with either the trivial grading or three distinct non-trivial Z 2 -gradings or by a Z 2 × Z 2 -grading. In this paper we prove that the variety of Jordan algebras generated by U J 2 endowed with any G-grading has the Specht property, i.e., every T G -ideal containing the graded identities of U J 2 is finitely based. Moreover, we prove an analogue result about the ordinary identities of A 1 , a suitable infinitely generated metabelian Jordan algebra defined in [27] .
Polynomial identities for the Jordan algebra of upper triangular matrices of order 2
2012
Abstract The associative algebras U T n ( K ) of the upper triangular matrices of order n play an important role in PI theory. Recently it was suggested that the Jordan algebra U J 2 ( K ) obtained by U T 2 ( K ) has an extremal behaviour with respect to its codimension growth. In this paper we study the polynomial identities of U J 2 ( K ) . We describe a basis of the identities of U J 2 ( K ) when the field K is infinite and of characteristic different from 2 and from 3. Moreover we give a description of all possible gradings on U J 2 ( K ) by the cyclic group Z 2 of order 2, and in each of the three gradings we find bases of the corresponding graded identities. Note that in the graded ca…
Growth of Differential Identities
2020
In this paper we study the growth of the differential identities of some algebras with derivations, i.e., associative algebras where a Lie algebra L (and its universal enveloping algebra U(L)) acts on them by derivations. In particular, we study in detail the differential identities and the cocharacter sequences of some algebras whose sequence of differential codimensions has polynomial growth. Moreover, we shall give a complete description of the differential identities of the algebra UT2 of 2 × 2 upper triangular matrices endowed with all possible action of a Lie algebra by derivations. Finally, we present the structure of the differential identities of the infinite dimensional Grassmann …
Triangular mass matrices of quarks and Cabibbo-Kobayashi-Maskawa mixing
1998
Every nonsingular fermion mass matrix, by an appropriate unitary transformation of right-chiral fields, is equivalent to a triangular matrix. Using the freedom in choosing bases of right-chiral fields in the minimal standard model, reduction to triangular form reduces the well-known ambiguities in reconstructing a mass matrix to trivial phase redefinitions. Furthermore, diagonalization of the quark mass sectors can be shifted to one charge sector only, without loosing the concise and economic triangular form. The corresponding effective triangular mass matrix is reconstructed, up to trivial phases, from the moduli of the CKM matrix elements, and vice versa, in a unique way. A new formula fo…
Riccati equation-based generalization of Dawson's integral function
2007
A new generalization of Dawson's integral function based on the link between a Riccati nonlinear differential equation and a second-order ordinary differential equation is reported. The MacLaurin expansion of this generalized function is built up and to this end an explicit formula for a generic cofactor of a triangular matrix is deduced.