Group gradings on the Lie and Jordan algebras of upper triangular matrices

Abstract

Let K be a field and let U T n = U T n (K) denote the associative algebra of upper triangular n × n matrices over K. The vector space of U T n can be given the structure of a Lie and of a Jordan algebra, respectively, by means of the new products: [a, b] = ab − ba, and a • b = ab + ba. We denote the corresponding Lie and Jordan algebra by U T − n and by U T + n , respectively. If G is a group, the G-gradings on U T n were described by Valenti and Zaicev (Arch Math 89(1):33-40, 2007); they proved that each grading on U T n is isomorphic to an elementary grading (that is every matrix unit is homogeneous). Also Di Vincenzo et al. (J Algebra 275(2):550-566, 2004) classified all elementary gradings on U T n. Here we study the gradings and the graded identities on U T − n and on U T +

Graded Lie-Rinehart Algebras

“…Taking into account now the third summand in (19), let be g , g ∈ [g] such that 0 = A g A g ⊂ A g g . If g g = 1 we have A (g ) −1 A g ⊂ A 1 , and so A (g ) −1 A g ⊂ A [g],1 .…”

“…Finally we consider the first summand A [g],1 A [g],1 in (19) and suppose there exist g , g ∈ [g] ∩ G such that ρ(L (g ) −1 )(A g ) + A (g ) −1 A g ρ(L (g ) −1 )(A g ) + A (g ) −1 A g = 0, so ρ(L (g ) −1 )(A g )ρ(L (g ) −1 )(A g ) + ρ(L (g ) −1 )(A g )(A (g ) −1 A g ) +(A (g ) −1 A g )ρ(L (g ) −1 )(A g ) + (A (g ) −1 A g )(A (g ) −1 A g ) = 0 (22) For the latter summand in Equation (22), in the case of g = (g ) −1 , the commutativity and associativity of A allow us to assert (A (g )…”

“…On the other hand, gradings appear elsewhere in the theory of Lie algebras, for example in the Cartan decomposition of a finite-dimensional complex semisimple Lie algebra (see for instance [1,6,10,15,16,19,23]). Also, graded modules have attracted the attention of many researchers in the last years (see [2,4,5,8,28,31]).…”

On the graded automorphisms of upper triangular matrix algebras

“…We explicitly exhibits all these groups for the upper triangular matrices. But first, we briefly recall all possible group gradings over the upper triangular matrices, as associative, Lie and Jordan algebras (see [14,2,11,12], or the survey [6]).…”