Russian Mathematical Surveys, 2023, Volume 78, Issue 4, Pages 785–787 DOI: https://doi.org/10.4213/rm10111e (Mi rm10111)

DOI: https://doi.org/10.4213/rm10111e

Bibliographic databases:

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

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

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]:

Since the operator $\mathcal{S}^{(3)}$ defined by (2) has the property

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

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:

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:

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 $q=1$.

Bibliography
1.	D. I. Gurevich, Leningrad Math. J., 2:4 (1991), 801–828
2.	D. Gurevich, P. Pyatov, and P. Saponov, St. Petersburg Math. J., 20:2 (2009), 213–253
3.	A. Isaev, O. Ogievetsky, and P. Pyatov, J. Phys. A, 32:9 (1999), L115–L121
4.	A. Polishchuk and L. Positselski, Quadratic algebras, Univ. Lecture Ser., 37, Amer. Math. Soc., Providence, RI, 2005, xii+159 pp.

Citation in format AMSBIB

\Bibitem{GurSapSok23}
\by D.~I.~Gurevich, P.~A.~Saponov, V.~V.~Sokolov
\paper On symmetrizers in quantum matrix algebras
\jour Russian Math. Surveys
\yr 2023
\vol 78
\issue 4
\pages 785--787
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\crossref{https://doi.org/10.4213/rm10111e}
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