Group gradings on the Lie and Jordan algebras of block-triangular matrices. (English) Zbl 1446.17041
The classifications of all group gradings on an algebra is a problem of significant importance in the theory of rings and algebras with an additional structure. The paper under review studies algebras of upper block-triangular matrices and their gradings. The authors study these as associative, Lie and Jordan algebras; in the Lie case they consider the traceless matrices only. Recall that the scalar matrices form the centre of the corresponding Lie algebra and it can be graded arbitrarily. Thus the authors classify the gradings on that algebra modulo its centre. Essentially they prove that every grading induces one on the full matrix algebra. Clearly there are gradings on the full matrix algebra that do not induce any grading on the block triangular matrices.
The authors classify all gradings by an abelian group \(G\) on the associative algebra of block triangular matrices, assuming the field algebraically closed. In this way they generalize theorems of Valenti and Zaicev concerning the case of algebras over an algebraically closed field of characteristic 0, see [A. Valenti and M. Zaicev, Can. Math. Bull. 55, No. 1, 208–213 (2012; Zbl 1244.16036)]. It turns out that the automorphism groups of the Lie algebra of the traceless block triangular matrices is isomorphic to that of the Jordan algebra of the block triangular matrices. This implies the authors can transfer the results from the Lie to the Jordan algebra case. Let me point out that while they consider gradings by abelian groups they classify gradings by any group by deducing that the support of the grading must be abelian. These results are far-reaching generalizations of their counterparts in the papers by P. Koshlukov and F. Y. Yasumura [Linear Algebra Appl. 534, 1–12 (2017; Zbl 1416.17023); J. Algebra 477, 294–311 (2017; Zbl 1390.17039)]. The latter two papers considered the Lie and Jordan algebras of upper triangular matrices only.
The paper is well written and contains a wealth of good ideas. The reader can find also several interesting comments and problems for research in the area.
The authors classify all gradings by an abelian group \(G\) on the associative algebra of block triangular matrices, assuming the field algebraically closed. In this way they generalize theorems of Valenti and Zaicev concerning the case of algebras over an algebraically closed field of characteristic 0, see [A. Valenti and M. Zaicev, Can. Math. Bull. 55, No. 1, 208–213 (2012; Zbl 1244.16036)]. It turns out that the automorphism groups of the Lie algebra of the traceless block triangular matrices is isomorphic to that of the Jordan algebra of the block triangular matrices. This implies the authors can transfer the results from the Lie to the Jordan algebra case. Let me point out that while they consider gradings by abelian groups they classify gradings by any group by deducing that the support of the grading must be abelian. These results are far-reaching generalizations of their counterparts in the papers by P. Koshlukov and F. Y. Yasumura [Linear Algebra Appl. 534, 1–12 (2017; Zbl 1416.17023); J. Algebra 477, 294–311 (2017; Zbl 1390.17039)]. The latter two papers considered the Lie and Jordan algebras of upper triangular matrices only.
The paper is well written and contains a wealth of good ideas. The reader can find also several interesting comments and problems for research in the area.
Reviewer: Plamen Koshlukov (Campinas)
MSC:
| 17B70 | Graded Lie (super)algebras |
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 17C99 | Jordan algebras (algebras, triples and pairs) |
References:
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