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Register a new userRingel-Hall algebra construction of quantum Borcherds-Bozec algebras
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Authors
Seok-Jin Kang
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We give the Ringel-Hall algebra construction of the positive half of quantum Borcherds-Bozec algebras as the generic composition algebras of quivers with loops.
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We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borcherds-Bozec algebras and quantum generalized Kac-Moody algebras.
Using the twisted denominator identity, we derive a closed form root multiplicity formula for all symmetrizable Borcherds-Bozec algebras and discuss its applications including the case of Monster Borcherds-Bozec algebra. In the second half of the paper, we provide the Schofield constuction of symmetric Borcherds-Bozec algebras.
Let $mathfrak{g}$ be a Borcherds-Bozec algebra, $U(mathfrak{g})$ be its universal enveloping algebra and $U_{q}(mathfrak{g})$ be the corresponding quantum Borcherds-Bozec algebra. We show that the classical limit of $U_{q}(mathfrak{g})$ is isomorphic to $U(mathfrak{g})$ as Hopf algebras. Thus $U_{q}(mathfrak{g})$ can be regarded as a quantum deformation of $U(mathfrak{g})$. We also give explicit formulas for the commutation relations among the generators of $U_{q}(mathfrak{g})$.
In this paper, we develop the theory of abstract crystals for quantum Borcherds-Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of ${B}(infty)$ and ${B}(lambda)$ as its application, where ${B}(infty)$ and ${B}(lambda)$ are the crystals of the negative half part of the quantum Borcherds-Bozec algebra $U_q(mathfrak g)$ and its irreducible highest weight module $V(lambda)$, respectively.
We investigate the fundamental properties of quantum Borcherds-Bozec algebras and their representations. Among others, we prove that the quantum Borcherds-Bozec algebras have a triangular decomposition and the category of integrable representations is semi-simple.
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