The 2 × 2 upper triangular matrix algebra and its generalized polynomial identities

Abstract

Let UT2 be the algebra of 2×2 upper triangular matrices over a field F of characteristic zero. Here we study the generalized polynomial identities of UT2, i.e., identical relations holding for UT2 regarded as UT2-algebra. We determine the generator of the TUTjavax.xml.bind.JAXBElement@5747df94-ideal of generalized polynomial identities of UT2 and compute the exact values of the corresponding sequence of generalized codimensions. Moreover, we give a complete description of the space of multilinear generalized identities in n variables in the language of Young diagrams through the representation theory of the symmetric group Sn. Finally, we prove that, unlike the ordinary case, the generalized variety of UT2-algebras generated by UT2 has no almost polynomial growth; nevertheless, we exhibit two distinct generalized varieties of almost polynomial growth.