| Russian Mathematical Surveys |
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Brief communications On symmetrizers in quantum matrix algebras D. I. Gurevicha, P. A. Saponovbc, V. V. Sokolova a Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow b National Research University Higher School of Economics c Institute for High Energy Physics, Protvino
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    1.In this note we deal with a particular class of quadratic algebras, which are called quantum matrix algebras. Recall that by a quadratic algebra one means a quotient algebra $A=T(V)/\langle J \rangle$, where $V$ is a vector space over the complex field $\mathbb{C}$, $\dim_{\mathbb{C}} V=N<\infty$, $T(V)=\bigoplus_{k\geqslant 0} V^{\otimes k}$ is the free tensor algebra, generated by $V$, and $\langle J \rangle$ is the ideal generated by the subspace $J\subset V^{\otimes 2}$. The algebraic properties of such algebras and of their inhomogeneous counterparts are intensively discussed in the mathematics literature (see, for instance, [4]). In this note we consider the problem of constructing some projections onto homogenous components of such algebras, which are analogues of usual symmetrizers. In the classical case such symmetrizers are defined via the flip $P$. Below, the role of $P$ is played by the so-called Hecke symmetries. Recall that an operator $R\colon V^{\otimes 2}\to V^{\otimes 2}$ is called a braiding if it is subject to the braid relation
$$
\begin{equation*}
R_{12} R_{23}\, R_{12}=R_{23} R_{12} R_{23},\qquad R_{12}=R\otimes I,\quad R_{23}=I \otimes R
\end{equation*}
\notag
$$
(here and below $I$ denotes the identity operator in $V$ or another vector space). If a braiding $R$ satisfies the complementary relation $(R-q I)(R+q^{-1} I)= 0$, where $q\in \mathbb{C}$, then $R$ is called a Hecke symmetry. In what follows we assume additionally that $q^k\ne 1$ for all $k\in \mathbb{Z}_+$. In [1] many Hecke symmetries were constructed and symmetrizers for the $R$- symmetric and $R$-skew-symmetric algebras of the space $V$ were defined. 2.Well-known examples of quadratic algebras related to $R$-matrices are the RTT- algebra and the reflection equation (RE) algebra. The first of them is defined by the system
$$
\begin{equation}
RT_1T_2=T_1T_2R, \qquad T_1=T\otimes I,\quad T_2=I\otimes T,
\end{equation}
\tag{1}
$$
where $T=\|t_i^j\|_{1\leqslant i,j \leqslant N}$ is an $N\times N$ matrix, whose entries generate the RTT-algebra. A method for constructing other matrix algebras was proposed in [3]. The main aim of this note is to present a method which hopefully enables one to construct symmetrizers onto homogeneous components of RTT-algebras and other matrix algebras. We illustrate this method by several low-dimensional examples. The defining system (1) can be written in the form $T_1 T_2={\mathcal{R}}(T_1 T_2)$, where ${\mathcal{R}}(T_1 T_2)=R^{-1} T_1 T_2 R$. If $R$ is a Hecke symmetry, then the operator ${\mathcal{R}}\colon W^{\otimes 2}\to W^{\otimes 2}$, where $W=\operatorname{span}(t_i^j)$, has three eigenvalues, $1$, $-q^2$, and $-q^{-2}$. Hence the operator $\mathcal{S}=2_q^{-2}({\mathcal{R}}+q^2I)({\mathcal{R}}+q^{-2}I)$, where $k_q \overset{\mathrm{def}}{=}(q^k-q^{-k})/(q-q^{-1})$, $k\in\mathbb{Z}$, is an idempotent, called a symmetrizer. It maps the space $W^{\otimes 2}$ onto its subspace of symmetric elements, that is, elements such that $\mathcal{S}(z)=z$. Thus, each element $z$ of the second homogeneous component $A^{(2)}$ of the quadratic algebra $A=T(W)/\langle J \rangle$ is equal to its symmetrized form $\mathcal{S}(z)$ modulo the ideal $\langle J \rangle$, where $J=\operatorname{Im}({\mathcal{R}}-I)$. The third-order symmetrizer $\mathcal{S}^{(3)}\colon W^{\otimes 3}\to W^{\otimes 3}$ projecting onto the homogeneous component $A^{(3)}\subset A$, was constructed in [2]:
$$
\begin{equation}
\begin{aligned} \, \mathcal{S}^{(3)}&:=\alpha \mathcal{S}_{12}\,\mathcal{S}_{23}\, \mathcal{S}_{12}\, \mathcal{S}_{23}\, \mathcal{S}_{12}+ \beta \mathcal{S}_{12}\,\mathcal{S}_{23}\, \mathcal{S}_{12}+ \gamma\mathcal{S}_{12} \nonumber \\ &\,=\alpha\mathcal{S}_{23}\, \mathcal{S}_{12}\, \mathcal{S}_{23}\, \mathcal{S}_{12}\,\mathcal{S}_{23}+ \beta \mathcal{S}_{23}\, \mathcal{S}_{12}\,\mathcal{S}_{23}+ \gamma\mathcal{S}_{23}, \end{aligned}
\end{equation}
\tag{2}
$$
where the coefficients $\alpha$, $\beta$, and $\gamma$ satisfy the equality $\alpha+\beta+\gamma=1$. Here and in what follows the subscripts of an operator indicate the indices of the components of the tensor product where the operator is applied. Since the operator $\mathcal{S}^{(3)}$ defined by (2) has the property
$$
\begin{equation*}
\mathcal{S}^{(3)}=\mathcal{S}_{12}\mathcal{S}^{(3)}= \mathcal{S}^{(3)} \mathcal{S}_{12}=\mathcal{S}_{23}\mathcal{S}^{(3)}= \mathcal{S}^{(3)} \mathcal{S}_{23},
\end{equation*}
\notag
$$
it can be written in the form $\mathcal{S}^{(3)}=p(\mathcal{S}_{12} \mathcal{S}_{23})= p(\mathcal{S}_{23}\mathcal{S}_{12})$, where $p$ is a polynomial of degree 3. The minimal polynomial of the operator $\mathcal{S}_{12} \mathcal{S}_{23}$ is of the form
$$
\begin{equation}
m_3(x)=x(x-1)\biggl(x-\frac{1}{2_q^2}\biggr) \biggl(x-\frac{(2_q^2-2)^2}{2_q^ {4}}\biggr).
\end{equation}
\tag{3}
$$
Consequently, the operator $p(\mathcal{S}_{12} \mathcal{S}_{23})/\kappa$, where $p(x)=m(x)(x-1)^{-1}$ and $\kappa=p(1)$, projects $W^{\otimes 3}$ onto the subspace corresponding to the eigenvalue 1. Note that formula (3) enables one to compute all coefficients in (2).
3.Now we construct the next symmetrizer $\mathcal{S}^{(4)}\colon W^{\otimes 4}\to W^{\otimes 4}$. To do this, first we compute the minimal polynomial $m_4(x)$ for the operator $\mathcal{S}^{(3)}_{123}\mathcal{S}^{(3)}_{234}\colon W^{\otimes 4}\to W^{\otimes 4}$. It is the fifth-order polynomial $m_4(x)=x(x-1)(x-\nu_1)(x-\nu_2)(x-\nu_3)$, where
$$
\begin{equation*}
\nu_1=\frac{1}{3_q^2}\,,\quad \nu_2=\frac{(2_q^2-2)^2}{4\cdot 3_q^2}\quad\text{and}\quad \nu_3=\frac{( 3_q^2-3)^2}{4\cdot 3_q^4}\,.
\end{equation*}
\notag
$$
Substituting the product $\mathcal{S}^{(3)}_{123}\, \mathcal{S}^{(3)}_{234}$ into the polynomial $p_4(x)=m_4(x)(x-1)^{-1}$, we obtain (after a proper normalization) the required symmetrizer $S^{(4)}$. It can be verified that the answer does not depend on the order of factors: $p_4(\mathcal{S}^{(3)}_{123}\mathcal{S}^{(3)}_{234})= p_4(\mathcal{S}^{(3)}_{234}\mathcal{S}^{(3)}_{123})$. Hopefully, a similar procedure enables one to find all higher symmetrizers. Provided that $\mathcal{S}^{(n)}$ is known, we define $\mathcal{S}^{(n+1)}$ as follows:
$$
\begin{equation*}
S^{(n+1)}_{1,2,\dots,n+1}= \kappa_{n+1}^{-1} p_{n+1} (S^{(n)}_{1,2,\dots,n} S^{(n)}_{2,3,\dots,n+1}),\qquad p_{n+1}(x)=m_{n+1}(x)(x-1)^{-1},
\end{equation*}
\notag
$$
where $m_{n+1}(x)$ is the minimal polynomial of the operator $S^{(n)}_{1,2,\dots,n}S^{(n)}_{2,3,\dots,n+1}$ and $\kappa_{n+1}\!=p_{n+1}(1)$ is a normalizing factor. This procedure is based on the conjecture that $1$ is a simple root of the polynomial $m_n(x)$ for any $n$. Presumably, the degree of $m_n(x)$ is $n+1$ for a generic $q$, but in the limit $q=1$ it becomes equal to $3$. This conjecture is justified by our computations for $n\kern-1pt=\kern-1pt5,6,7$. The roots of $m_5(x)$ are $0$, $1$, and the following numbers:
$$
\begin{equation*}
\nu_1=\frac{1}{4_q^2}\,, \quad \nu_2=\frac{(2\cdot 2_q^2-5)^2}{9\cdot 2_q^4}\,, \quad \nu_3=\frac{(2\cdot 2_q^2-5)^2}{9\cdot 4_q^2}\quad\text{and}\quad \nu_4=\frac{(4_q^2-4)^2}{9\cdot 4_q^4}\,.
\end{equation*}
\notag
$$
It would be interesting to find a regular pattern in the series of the polynomials $m_n(x)$. Note that the existence of a symmetrizer $S^{(n)}$ entails the following claim. If a Hecke symmetry $R=R(q)$ is an analytic matrix-function of $q$ in a neighbourhood of $q=1$, then in some neighbourhood of $q=1$ the dimensions of the homogenous components $A^{(n)}$ are equal to those for $q=1$. 
Citation in format AMSBIB
\Bibitem{GurSapSok23} |
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