The index of Lie poset algebras

The Index and Spectrum of Lie Poset Algebras of Types B, C, and D

“…Remark 4. Restricting g(P) to trace-zero matrices results in a subalgebra of A n−1 = sl(n), which we call a type-A Lie poset algebra (see [2,6,7]). In [2], the authors show that type-A Lie poset algebras are equivalent to subalgebras of sl(n) lying between the Borel subalgebra of sl(n) consisting of all upper-triangular matrices and its Cartan subalgebra of diagonal matrices.…”

“…In [2], the authors show that type-A Lie poset algebras are equivalent to subalgebras of sl(n) lying between the Borel subalgebra of sl(n) consisting of all upper-triangular matrices and its Cartan subalgebra of diagonal matrices. Coll and Mayers [6,7] initiated an investigation into the index and spectral theories of type-A Lie poset algebras.…”

“…In this section, we provide definitions for posets of types B, C, and D (cf. [6]). These posets encode matrix forms that define Lie poset algebras of types B, C, and D.…”

“…In the type-A setting, the height of a poset P is defined to be one less than the largest cardinality of a chain. If P is a connected, height-one (type-A) poset, then the first two authors recently established that the index of the associated type-A Lie poset algebra g A (P) is given by the following closed-form expression (see Theorem 4, [6]):…”

“…The development of combinatorial index formulas for certain families of Lie algebras is of current interest. Families for which such formulas have been found include seaweed algebras and type-A Lie poset algebras (see [4,5,6,8,9,12,17,19,20,21]). In this article, we consider the analogues of type-A Lie poset algebras in the other classical types: B k = so(2k + 1), C k = sp(2k), and D k = so(2k); such algebras are called type-B, C, and D Lie poset algebras, respectively.…”