Identidades e cocaracteres da *-superálgebra M2,1(F) com involução transposta - Juan Antonio Pacheco Cruz (2022)
Let F be a field of characteristic zero. Considering that the T-ideal of the algebra of matrices M3(F) is still unknown, some authors study such algebra endowed with additional structures. This thesis was inspired by the results of La Mattina (in [16]) about the graded identities of the superalgebra M2,1(F) and also by the results of D’Amour and Racine (in [4]) about the ∗-identities of M3(F) with transpose involution t. Here, we are devoted to the study of the so-called (Z2, ∗)-identities of the ∗-superalgebra (M2,1(F), t) and determine all such identities of degree up to 3. Furthermore, we study the decomposition of the so-called ∗-graded cocharacter of (M2,1(F), t), motivated by the papers [1], [16] and [2], with respect to the non-zero multiplicities in the decompositions of the Sn-cocharacter of M3(F), of the graded cocharacter of the superalgebra M2,1(F) and of the ∗-cocharacter of the ∗-algebra (M3(F), t), respectively. We present necessary and sufficient conditions for having non-zero multiplicity of an irreducible character to appear in the nth ∗-graded cocharacter of (M2,1(F), t).