Hermitian representations of a TKK algebra
Extended affine Lie algebras are generalizations of finite-dimensional simple Lie algebras and affine Kac-Moody algebras. It was systematically studied in [\textit{B. N. Allison} et al., Extended affine Lie algebras and their root systems. Providence, RI: American Mathematical Society (AMS) (1997; Zbl 0879.17012)]. The core of an extended affine Lie algebra is the subalgebra generated by non-isotropic root spaces. The core of an extended affine Lie algebra of type \(A_1\) is a central extension of Tits-Kantor-Koecher Lie algebra \(\mathfrak{G}(\mathfrak{J}(\Lambda))\) (see [\textit{N. Jacobson}, Structure and representations of Jordan algebras. American Mathematical Society (AMS), Providence, RI (1968; Zbl 0218.17010)] and Chapter III of [\textit{B. N. Allison} et al., Extended affine Lie algebras and their root systems. Providence, RI: American Mathematical Society (AMS) (1997; Zbl 0879.17012)]). Here \(\mathfrak{J}(\Lambda)\) is one of Jordan tori \(\mathbb{C}^+_Q\), hermitian torus, Clifford torus, or Albert torus. The paper constructs the free fields representation for \(\mathfrak{G}(\mathfrak{J}(S))\), where \(\mathfrak{J}(S)\) is the smallest Clifford Jordan torus. Since the non-isotropic roots of \(\mathfrak{G}(\mathfrak{J}(S))\) form a semi-lattice in \(\mathbb{Z}_2\), the module of the given construction has less variables than the one given in [\textit{X. Mao} and \textit{S. Tan}, ``Wakimoto representation for the Tits-Kantor-Koecher Lie algebra'' (Chinese), Chin. Ann. Math. Ser. A 28, No. 4, 329--338 (2007)]. The paper also gives the conditions for the module being irreducibility, construct a hermitian form on the module and determine when the representation is unitary.
scientific article; zbMATH DE number 7396013
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