TY - JOUR

T1 - Commutator relations and structure constants for rank 2 Kac–Moody algebras

AU - Carbone, Lisa

AU - Kownacki, Matt

AU - Murray, Scott H.

AU - Srinivasan, Sowmya

N1 - Funding Information:
The research of the first author is partly supported by Simons Foundation Collaboration Grant, no. 422182.

PY - 2021/1/15

Y1 - 2021/1/15

N2 - We completely determine the structure constants between real root vectors in a rank 2 Kac–Moody algebra g. Our description is computationally efficient, even in the rank 2 hyperbolic case where the coefficients of roots on the root lattice grow exponentially with height. Our approach is to extend Carter's method of finding structure constants from those on extraspecial pairs to the rank 2 Kac–Moody case. We also determine all commutator relations involving only real root vectors in all rank 2 Kac-Moody algebras. The generalized Cartan matrix of g is of the form H(a,b)=(2−b−a2) where a,b∈Z and ab≥4. If ab=4, then g is of affine type. If ab>4, then g is of hyperbolic type. Explicit knowledge of the root strings is needed, as well as a characterization of the pairs of real roots whose sums are real. We prove that if a and b are both greater than one, then no sum of real roots can be a real root. We determine the root strings between real roots β,γ in H(a,1), a≥5 and we determine the sets (Z≥0α+Z≥0β)∩Δre(H(a,b)). One of our tools is a characterization of the root subsystems generated by a subset of roots. We classify these subsystems in rank 2 Kac–Moody root systems. We prove that every rank two infinite root system contains an infinite family of non-isomorphic symmetric rank 2 hyperbolic root subsystems H(k,k) for certain k≥3, generated by either two short or two long simple roots. We also prove that a non-symmetric hyperbolic root systems H(a,b) with a≠b and ab>5 also contains an infinite family of non-isomorphic non-symmetric rank 2 hyperbolic root subsystems H(aℓ,bℓ), for certain positive integers ℓ.

AB - We completely determine the structure constants between real root vectors in a rank 2 Kac–Moody algebra g. Our description is computationally efficient, even in the rank 2 hyperbolic case where the coefficients of roots on the root lattice grow exponentially with height. Our approach is to extend Carter's method of finding structure constants from those on extraspecial pairs to the rank 2 Kac–Moody case. We also determine all commutator relations involving only real root vectors in all rank 2 Kac-Moody algebras. The generalized Cartan matrix of g is of the form H(a,b)=(2−b−a2) where a,b∈Z and ab≥4. If ab=4, then g is of affine type. If ab>4, then g is of hyperbolic type. Explicit knowledge of the root strings is needed, as well as a characterization of the pairs of real roots whose sums are real. We prove that if a and b are both greater than one, then no sum of real roots can be a real root. We determine the root strings between real roots β,γ in H(a,1), a≥5 and we determine the sets (Z≥0α+Z≥0β)∩Δre(H(a,b)). One of our tools is a characterization of the root subsystems generated by a subset of roots. We classify these subsystems in rank 2 Kac–Moody root systems. We prove that every rank two infinite root system contains an infinite family of non-isomorphic symmetric rank 2 hyperbolic root subsystems H(k,k) for certain k≥3, generated by either two short or two long simple roots. We also prove that a non-symmetric hyperbolic root systems H(a,b) with a≠b and ab>5 also contains an infinite family of non-isomorphic non-symmetric rank 2 hyperbolic root subsystems H(aℓ,bℓ), for certain positive integers ℓ.

KW - Commutator relations

KW - Kac-Moody algebra

KW - Structure constants

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U2 - 10.1016/j.jalgebra.2020.08.023

DO - 10.1016/j.jalgebra.2020.08.023

M3 - Article

AN - SCOPUS:85092093248

VL - 566

SP - 443

EP - 476

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -