The centralizer construction and Yangian-type algebras. (English) Zbl 1536.17021
This paper investigates a certain generalization of the centralizer construction [the author, Sov. Math., Dokl. 36, No. 3, 569–573 (1988; Zbl 0662.22016); translation from Dokl. Akad. Nauk SSSR 297, 1050–1054 (1987); J. Sov. Math. 47, No. 2, 2466–2473 (1989; Zbl 0691.17005); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 164, 142–150 (1987); Adv. Sov. Math. 2, 1–66 (1991; Zbl 0739.22015)]. More specifically, for any integer \(L\ge 2\), consider the Lie algebras \[ \mathfrak{a}(N)=\mathfrak{gl}(N,\mathbb{C})^{\oplus L}\quad\text{and}\quad \mathfrak{b}(N)=\mathop{\textrm{diag}}(\mathfrak{gl}_d(N,\mathbb{C})), \] where \(\mathfrak{gl}_d(N,\mathbb{C})\cong \mathfrak{gl} (N-d,\mathbb{C})\) is the Lie subalgebra of \(\mathfrak{gl}(N,\mathbb{C})\) which consists of all matrices of the form \( \begin{pmatrix} 0&0\\ 0&X \end{pmatrix} \) such that \(X\) is a \((N-d)\times (N-d)\) complex matrix. Let \[ A_{d,L}(N)=U(\mathfrak{a}(N))^{\mathfrak{b}(N)} \quad\text{with }d=0,\ldots ,N \] be the centralizers of \(\mathfrak{b}(N)\) in the universal enveloping algebra \(U(\mathfrak{a}(N))\). The author studies the projective limit filtered algebras \[ A_{d,L}=\lim_{\leftarrow}(A_{d,L}(N),\pi_{N,N-1}),\quad N\to\infty, \] where \[ \pi_{N,N-1}\colon A_{d,L}(N)\to A_{d,L}(N-1) \quad\text{with }N>d \] are certain algebra morphisms. In particular, the algebra \(A_{0,L}\) obtained in the case \(d=0\) embeds into \(A_{d,L}\).
The main results of the manuscript are as follows. First, the author proves that, for all \(d\ge 1\) and \(L\ge 2\), the algebra \(A_{d,L}\) possesses a subalgebra \(Y_{d,L}\) such that the multiplication map \(A_{0,L}\otimes Y_{d,L}\to A_{d,L}\) is a vector space isomorphism. It is worth noting that the case \(L=1\) of this result is well-known (see the author, Zbl 0739.22015) and, in addition, that the corresponding algebras \(Y_{d,1}\) are exactly the Yangians \(Y_{d,1}=Y(\mathfrak{gl}(d,\mathbb{C}))\). On the other hand, the case \(L\ge 2\), studied in this paper, leads to a new family \(Y_{d,L}\) of Yangian-type algebras. Finally, the author shows that these algebras possess a presentation by countably many generators subject to quadratic-linear commutation relations. However, it is not known whether, as with the ordinary Yangians, they possess a RTT-type presentation as well.
The main results of the manuscript are as follows. First, the author proves that, for all \(d\ge 1\) and \(L\ge 2\), the algebra \(A_{d,L}\) possesses a subalgebra \(Y_{d,L}\) such that the multiplication map \(A_{0,L}\otimes Y_{d,L}\to A_{d,L}\) is a vector space isomorphism. It is worth noting that the case \(L=1\) of this result is well-known (see the author, Zbl 0739.22015) and, in addition, that the corresponding algebras \(Y_{d,1}\) are exactly the Yangians \(Y_{d,1}=Y(\mathfrak{gl}(d,\mathbb{C}))\). On the other hand, the case \(L\ge 2\), studied in this paper, leads to a new family \(Y_{d,L}\) of Yangian-type algebras. Finally, the author shows that these algebras possess a presentation by countably many generators subject to quadratic-linear commutation relations. However, it is not known whether, as with the ordinary Yangians, they possess a RTT-type presentation as well.
Reviewer: Slaven Kozic (Zagreb)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B63 | Poisson algebras |
| 17B35 | Universal enveloping (super)algebras |
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