| Russian Mathematical Surveys |
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Brief communications Cyclic Frobenius algebras V. M. Buchstaberab, A. V. Mikhailovc a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow b Lomonosov Moscow State University c University of Leeds, Leeds, UK
Russian version:
Uspekhi Matematicheskikh Nauk, 2023, Volume 78, Issue 1(469), DOI: https://doi.org/10.4213/rm10096 
Bibliographic databases:
    Let $\mathcal{A}$ be an associative $\mathbb{C}$-algebra with unit element 1 and $\mathcal{M}$ be a $\mathbb{C}$-linear space, $\dim\mathcal{M}\geqslant 1$. Definition 1. A cyclic Frobenius algebra (a $\operatorname{CF}$-algebra) $\mathcal{A}$ is by definition an algebra $\mathcal{A}$ with $\mathbb{C}$-bilinear skew-symmetric form $\Phi(\,\cdot\,{,}\,\cdot\,) \colon \mathcal{A}\otimes_\mathbb{C} \mathcal{A}\to \mathcal{M}$ such that $ \Phi(A,BC)+\Phi(B,CA)+\Phi(C,AB)=0$, $A,B,C \in\mathcal{A}$. Examples: 1) $\mathcal{M}=\mathcal{A}$ and $\Phi(A,B)=AB-BA=[A,B]$; 2) $\mathcal{A}$ is a commutative algebra with Poisson bracket $\{\,\cdot\,{,}\,\cdot\,\}$, $\mathcal{M}=\mathcal{A}$, and $\Phi(A,B)=\{A,B\}$; 3) cyclic Frobenius algebras with $\mathcal{M}$ a field can be obtained using the constructions of algebras from [1] and [5]. Consider the $\mathbb{C}$-linear subspace $\operatorname{Span}$ of $ \mathcal{A}$ spanned by all commutators $[A,B]\in \mathcal{A}$, and consider the projection $\pi\colon \mathcal{A}\to \mathcal{A}/\operatorname{Span}$. Set $A\approx B$ if $A-B\in\operatorname{Span}$. Lemma 1. Let $\varphi(\,\cdot\,{,}\,\cdot\,)\colon \mathcal{A}\otimes \mathcal{A}\to \mathcal{A}$ be a $\mathbb{C}$-bilinear skew-symmetric form, and let $\varphi(A,BC)\approx\varphi(A,B)C+B\varphi(A,C)$. Then $\mathcal{A}$ is a cyclic Forbenius algebra for $\mathcal{M}=\mathcal{A}/\operatorname{Span}$ and $\Phi=\pi\varphi$. Set $\mathcal{K}=\{B\in\mathcal{A}\colon\Phi(A,B)=0 \text{ for all } A\in\mathcal{A}\}$. It follows from Definition 1 that $\mathcal{K}$ is a subring of $\mathcal{A}$, $1\in \mathcal{K}$, and $\Phi(A,bC)=\Phi(Ab,C)$ for all $A,C \in\mathcal{A}$ and $b\in\mathcal{K}$. Let $\mathfrak{A}=\mathbb{C}\langle u_0,u_1,\ldots\rangle= \textstyle\bigoplus_{k=0}^\infty\mathfrak{A}_k$ be a free associative graded algebra with differentiation operator $D$, let $|u_k|=k+2$, and let $D(u_k)=u_{k+1}$, $k=0,1,\dots$ . Consider the algebra $\mathfrak{A}^D= \{A=\sum_{i\leqslant m}a_iD^i,\, a_m\ne 0,\, m\in \mathbb{Z},\, a_i\in \mathfrak{A}_{|a_m|+m-i}\}$, where $[D,u_k]=u_{k+1}$ and $[D^{-1},u_k]=\sum_{i\geqslant 1}(-1)^i u_{k+i}D^{-i-1}$. Let $A_++A_-=A \in \mathfrak{A}^D$, where $A_+= \sum_{0\leqslant i\leqslant m} a_iD^{i}$ for $m\geqslant 0$ and $A_+=0$ for $m<0$. Then we have $\operatorname{res}[D,A]=D(\operatorname{res}A)$, where $\operatorname{res} A=a_{-1}$. Let $\sigma(\,\cdot\,{,}\,\cdot\,)\colon \mathfrak{A}^D\otimes\mathfrak{A}^D \to \mathfrak{A}$ be a homogeneous $\mathbb{C}$-bilinear skew-symmetric form satisfying $\sigma(A,B)|=|A|+|B|$ and defined by
$$
\begin{equation*}
\sigma(aD^n,bD^m)= \frac{1}{2}\begin{pmatrix} n\\ n+m+1\end{pmatrix} \sum_{s=0}^{n+m}\!(-1)^s (a^{(s)}b^{(n+m-s)}\!+b^{(n+m-s)}a^{(s)}),
\end{equation*}
\notag
$$
for $n+m \geqslant 0$, $nm<0$, and by $\sigma(aD^n,bD^m)=0$ otherwise.
Lemma 2. For any $A,B \in \mathfrak{A}^D$ the equality $D(\sigma(A,B))=\operatorname{res}[A,B]-\Delta(A,B)$ holds, where for $n+m \geqslant -1$ and $\Delta(aD^n,bD^m)=0$ otherwise. Lemma 3. Under the canonical projection $\pi\colon \mathfrak{A}\to \mathfrak{A}/\operatorname{Span}=\mathcal{M}=\bigoplus_{k=0}^\infty\mathcal{M}_k$ the operator $D$ induces a monomorphism $\overline{D}\colon \mathcal{M}_k \to \mathcal{M}_{k+1}$, $k>0$. Theorem 1. The algebra $\mathfrak{A}^D$ is a cyclic Forbenius algebra with form $\Phi=\pi\sigma\colon \mathfrak{A}^D\otimes\mathfrak{A}^D\to \mathcal{M}$ such that $\mathcal{K}=\mathfrak{A}$ and $\Phi(D^n,D^{-n})=n$, $n\in \mathbb{Z}$. Corollary 1. The form $\Phi$ is non-degenerate on the set $\{D^n,\, n\in \mathbb{Z}\}$ of generators of the free left $\mathfrak{A}$-module $\mathfrak{A}^D$. Set $L=D^2-u$ and $\mathcal{L}=D+ \sum_{i\geqslant1} I_{i}D^{-i}$, $I_{i}\in \mathfrak{A}_{i+1}$, where $\mathcal{L}^2=L$. Then we obtain the sequence of series $\mathcal{L}^{2k-1}$, $k\in\mathbb{N}$. Set
$$
\begin{equation*}
\sigma_{2k-1,2n-1}=\sigma(\mathcal{L}_+^{2k-1},\mathcal{L}^{2n-1})\in\mathfrak{A}_{2n+2k-2} \ \ \text{and}\ \ \rho_{2n}=\sigma_{1,2n-1}=\operatorname{res}\mathcal{L}^{2n-1},\ \ n>0.
\end{equation*}
\notag
$$
From the properties of $\sigma(\,\cdot\,{,}\,\cdot\,)$ we obtain the equality $\sigma_{2k-1,2n-1}=\sigma_{2n-1,2k-1}$ for $k,n\in\mathbb{N}$.
We introduce differentiations $D_{2k-1}$, $k\in\mathbb{N}$, of the ring $\mathfrak{A}$ by setting
$$
\begin{equation*}
D_1=D,\quad [D,D_{2k-1}]=0,\quad D_{2k-1}(u)=-2D(\rho_{2k}),\quad k\in\mathbb{N}.
\end{equation*}
\notag
$$
Let $u=u(t_1,t_3,\dots)$. Set $\partial_{t_{2k-1}}(u)=D_{2k-1}(u)$. Theorem 2. The system of equations $\partial_{t_{2k-1}}(u)=-2D(\rho_{2k}(u))$, $k\in \mathbb{N}$, where $\rho_2(u)=-u/2$, coincides with the Korteweg–de Vries (KdV) hierarchy on $\mathfrak{A}$: Following [3], for $N\in \mathbb{N}$ set $F_{2N+2}(u)=\rho_{2N+2}+\sum_{k=0}^{N-1}\alpha_{2(N-k+1)}\rho_{2k}$, where $\rho_0= 1$ and $\alpha_4,\dots,\alpha_{2N+2}$ are free parameters. The equation $F_{2N+2}(u)=0$ is called Novikov’s $N$-equation. Let $J(F)$ be the two-sided $D$-differential ideal in $\mathfrak{A}$ generated by $F_{2N+2}(u)$. Since $2^{2k+1}\rho_{2k+2}=u_{2k}+f(u_{2k-2},\dots,u)$ and $u_{k+1}=D^k(u)$, on the quotient algebra $\mathfrak{A}/J(F)=\mathbb{C}\langle u,\dots,u_{2N-1}\rangle$ the KdV hierarchy (see Theorem 2) reduces to Novikov’s $N$-hierarchy, where the first system represents Novikov’s $N$-equation in the form $D(u_k)=u_{k+1}$ for $0\leqslant k< 2N-1$ and $D(u_{2N-1})=-f(u_{2k-2},\dots,u)$. It was shown in [3] that, in terms of the form $\Phi$ (see Theorem 1), the polynomials
$$
\begin{equation}
H_{2n+1,2N+1}=\sigma_{2n+1,2N+1}+ \sum_{k=1}^{N-1}\alpha_{2(N-k+1)}\sigma_{2n+1,2k-1},\qquad n=1,\dots,N,
\end{equation}
\tag{1}
$$
define the first integrals $\widehat{H}_{2n+1,2N+1}=\pi(H_{2n+1,2N+1})$ of this hierarchy.
Theorem 3. In the quantum case Novikov’s $N$-hierarchy (see [3]) can be written as consistent systems of Heisenberg’s equation. The polynomials (1) are commuting quantum Hamiltonians of this hierarchy, which are selfadjoint in the case when the parameters $\alpha_4,\dots,\alpha_{2N+2}$ take real values. The authors are grateful to S. P. Nivikov and V. N. Rubtsov for stimulating discussions of the results presented in this note.
Citation in format AMSBIB
\Bibitem{BucMik23} |
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