Polynomial rings as representations of Lie Algebras of vector space endomorphisms

online | | 14:30

Ommolbanin Behzad

University of Isfahan, Isfahan, Iran

Any vector space is a representation of the Lie algebra of its own endomorphism. By means of this basic fact, in a recent work joint with A.Contiero and D. Martins, we studied how to uniformly represent Lie algebras of vector space endomorphisms on exterior algebras: Restricting the representation to the irreducible sub-representations given by th exterior power, and using the isomorphisms of exterior power with polynomial rings, one finally obtains the finite type version of the celebrated DJKM bosonic vertex operator representation of $gl_\infty(\mathbb{Q})$: The basic tool we use is the notion of Schubert derivation introduced by Gatto since 2005.

[1] O. Behzad, Hasse-Schmidt Derivations and Vertex Operators on Exterior Algebras. Ph.D. Thesis, IASBS, Zanjan, 2021;
[2] O. Behzad, A. Contiero, D. Martins, Vertex Operators Representation of Lie Algebras of Matrices, 2021, Available at ArXiv:2108.12895.pdf

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