On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras
Let \((E,\Pi,p)\) be a triple consisting of a finite dimensional complex space \(E\) with a non-degenerate symmetric bilinear form \((\;,\;)\), a linearly independent subset \(\Pi=\{\alpha_0,\dots, \alpha_n\}\) in \(E\) and a function \(p:\Pi\to \{0,1\}\). There is a Lie superalgebra \(\widetilde L=\widetilde L(E,\Pi,p)\) which is defined by the set of generators \(H=E^*, E_i, F_i\), \(0\leq i\leq n-1\) subject to the defining relations \[ [H,H']=0,\quad [H,E_i]=\alpha_i(H)E_i,\quad [H,F_i]=-\alpha_i(H)F_i,\quad [E_i,F_j]= \delta_{ij}H_{\alpha_i}. \] Here \(H_{\alpha_i}\in E^*\) is the unique element such that \((x,H_{\alpha_i})=(x,\alpha_i)\). There exists the unique maximal ideal in \(\widetilde L\), whose intersection with \(H^*\) is trivial. Then \(L(E,\Pi,p)\) is the factor by this ideal. If \(\mathfrak g\) is a Lie superalgebra with an automorphism \(\tau\) of a finite order \(r\), then one can define a Lie superalgebra \[ L(\mathfrak g,\tau)= \oplus_{i=0}^{r-1} \Bigl(\oplus_{m\in \mathbb Z} \mathfrak g_n^{\tau} \otimes t^{mr+n}\Bigr) \oplus \mathbb C c\oplus \mathbb Cd \] with central elements \(c,d\) of degree 0, where \[ \mathfrak g_n^{\tau}= \biggl\{X\in \mathfrak g\mid \tau(X)= \exp\frac {2\pi n }r X\biggr\}. \] The main result of the paper is a classification of defining relations for the generators \(H=E^*, E_i, F_i\), \(0\leq i\leq n-1,\) of the affine quantized universal enveloping algebra \(U_q(L(E,\Pi,p))\) with the only exception for \(U_q(A(m,m)^{(2)}), U_q(A(m,m)^{(4)})\). Some of these results were announced in the author's paper [Proc. Japan Acad., Ser. A 70, 31--36 (1994; Zbl 0987.17008), see also Publ. Res. Inst. Math. Sci. 30, 15--87 (1994; Zbl 0821.17005)].
scientific article; zbMATH DE number 1439854
scientific article