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Explicit expressions for Šapovalov elements in Type A by Ian M. Musson is a Mathematics article available to read on EtoBox.
What is Explicit expressions for Šapovalov elements in Type A about?
We give explicit expressions for Šapovalov elements in Type A Lie algebras and superalgebras. Explicit expressions were already given in [11] Section 9, using non-commutative determinants, and in fact our first main results, Theorems 2.3 and 2.6 can be viewed as complete expansions of these determinants. But we give new proofs, which seem easier because they avoid induction and cofactor expansion. We also describe Šapovalov elements for gl ( m , n ) with respect to an arbitrary Borel subalgebra in Theorem 3.7, and interpret Šapovalov elements in Type A as determinants of Hessenberg matrices in Theorems 4.5 and 4.9. The exact form of the explicit expressions depends on an ordering on the set of positive roots, and Hessenberg matrices are useful in changing the ordering. Having explicit expressions for Šapovalov elements allows us to give easy proofs of several results in representation theory, see Section 6 and Subsection 4.6.
Who reads Explicit expressions for Šapovalov elements in Type A?
It is typically read by researchers, students, and practitioners in Mathematics.
- Author
- Ian M. Musson
- Publisher
- Elsevier BV
- Published
- 2023
- Language
- EN
- Field
- Mathematics (Physical Sciences)