Quantum Symplectic Monoids

Let ${R}$ be a noetherian integral domain and ${q}$$\in R$ an invertible element. Let ${V}$ be a free $R$-module of rank $n=2m$. Fix a basis $\{v_{1}, \ldots , v_{n}\}$    and let $e_{ij}$ denote the corresponding basis of matrix units for ${{\cal E}}$$:={\rm End}_{R}{(V)}$. We will define two endomorphisms ${\beta}$ and ${\gamma}$ on $V\otimes V$ identifying ${\rm End}_{R}{(V\otimes V)}$ with ${\cal E}\otimes{\cal E}$ (we write simply $\otimes$ instead of $\otimes_{R}$ if no ambiguity can arise). Some additional notation is needed. We set

$$(\rho_1, \ldots , \rho_n)=(m, m-1, \ldots, 1, -1,\ldots ,-(m-1), -m)$$   $$\mbox{\index{${\rho_i}$}}$$$$$$

and ${\epsilon_i}$$:=sign(\rho_i).$ Further, ${i'}$$:=n-i+1$ defines an involution on ${\underline{n}}$$:=\{1,\ldots , n\}$. Thus

$$(1',2', \ldots , n')=(n,n-1, \ldots , 1).$$

The following definition is taken from [Ha2, Equation (4.3),(4.5)] (resp. [Ha1, section 5]) using the transformation $\beta= q^2\beta_{q^{-1}}(C_m)$ and $\gamma=\iota_q$.

$$\beta:=\sum_{1\leq i\leq n} (q^2e_{ii} \otimes e_{ii} +e_{ii'} \otimes e_{i'i}) + q\sum_{1\leq i\neq j,j'\leq n} e_{ij} \otimes e_{ji} +$$

$$+(q^2-1)\sum_{1\leq j < i\leq n} (e_{ii} \otimes e_{jj} -q^{\rho_i -\rho_j}\epsilon_i\epsilon_j e_{ij'} \otimes e_{i'j}),$$

and

$$\gamma: =\sum_{1\leq i,j\leq n} q^{\rho_i -\rho_j}\epsilon_i\epsilon_j e_{ij'} \otimes e_{i'j},$$

There are slightly more general versions of these endomorphisms involving additional parameters. We may omit them without loss of generality (see [O1, Satz 2.5.8]). The operators $\beta$ and $\gamma$ are related to each other by the equation (cf. [Ha2, Equation (4.4)])

$$(q^2-1)(\gamma -{\rm id}_{V^{\otimes 2}})=q^2\beta^{-1} -\beta$$ (1)

For $r\in {\mathbb{N}}\,$ write $\underline{r}:=\{1, \ldots , r\}$. A multi-index is a map ${{\bf i}}$$:\underline{r}\rightarrow \underline{n}$ frequently denoted as an $r$-tuple ${\bf i}=(i_1, \ldots , i_r)$ where $i_j\in \underline{n}$. The set of all such multi-indices will be denoted by ${I(n,r)}$. We define

$${v_{{\bf i}}}$$$$\mbox{\index{${v_{{\bf i}}}$}}$$$$:=v_{i_1}\otimes v_{i_2} \otimes \ldots \otimes v_{i_r} \in \und... ...otimes V}_{ r\mbox{-times}}=: {V^{\otimes r}}\mbox{\index{${V^{\otimes r}}$}}.$$

An endomorphism $\mu$ of $V^{\otimes r}$ may be given by its coefficients $\mu_{{\bf i}{\bf j}}$ with respect to the basis $\{ v_{{\bf i}}\vert\; {\bf i} \in I(n,r)\}$ of $V^{\otimes r}$, that is

$$\mu (v_{{\bf j}}) = \sum_{{\bf i}\in I(n,r)}\mu_{{\bf i}{\bf j}}v_{{\bf i}}.$$

Let ${F_{R}(n)}$$:=R\langle X_{1 1},X_{1 2}, \ldots , X_{n n}\rangle$ be the free algebra generated by the $n^2$ symbols $X_{i j}$ for $i,j \in \underline{n}$. This is a graded algebra; an $R$-basis of the $r$-th homogeneous part $F_{R}(n,r)$ is the set

$$\{ X_{{\bf i} {\bf j}}:=X_{i_1 j_1}\cdots X_{i_2 j_2}\cdots X_{i_r j_r}\vert\; \; {\bf i},{\bf j} \in I(n,r)\}.$$

To simplify notation we introduce a new convention to write down frequently used elements of $F_{R}(n)$ and its quotients in a convenient way. For an endomorphism $\mu$ on $V^{\otimes r}$ we write

$$\mu \wr X_{{\bf i} {\bf j}} :=\sum_{{\bf k}\in I(n,r)} \mbox{\index{${-\wr -}$}} \mu_{{\bf i}{\bf k}}X_{{\bf k} {\bf j}} \; \; \; \; X_{{\bf i} {\bf j}}\wr \mu :=\sum_{{\bf k}\in I(n,r)} X_{{\bf i} {\bf k}}\mu_{{\bf k}{\bf j}}.$$ (2)

This definition can be linearly extended to all of $F_{R}(n,r)$. The following rules are easily checked.

$$\begin{array}{rl} 1\wr X_{{\bf i} {\bf j}} & = X_{{\bf i} {\... ...})\wr \nu & = \nu\wr (X_{{\bf i} {\bf j}}\wr \nu ) \end{array}$$ (3)

We will denote the residue classes of $X_{i j}$ in any quotient of $F_{R}(n)$ by ${x_{i j}}$. The residue class ${x_{{\bf i} {\bf j}}}$ of ${X_{{\bf i} {\bf j}}}$ then clearly has a similar expression in the $x_{i j}$ as the $X_{{\bf i} {\bf j}}$ do in the $X_{i j}$. The above introduced convention will be used for $x_{{\bf i} {\bf j}}$ accordingly.

The object of our investigations is given by the following definition:

$${A^{{\rm s}}_{R,q}(n)}$$$$\mbox{\index{${A^{{\rm s}}_{R,q}(n)}$}}$$$$:= F_{R}(n)/\left< \beta\wr X_{{\bf i} {\bf j}}-X_{{\bf i} {\bf ... ...}}-X_{{\bf i} {\bf j}}\wr \gamma \vert\; \; {\bf i},{\bf j} \in I(n,2)\right>.$$

Here the brackets $\langle\rangle$ denote the ideal generated by the enclosed elements and $\beta, \gamma$ are the endomorphisms on $V\otimes V$ defined above. Since this ideal in the definition is homogeneous, the algebra $A^{{\rm s}}_{R,q}(n)=\bigoplus_{r\in {\mathbb{N}}\,_0} A^{{\rm s}}_{R,q}(n,r)$ is again graded. Here, ${ A^{{\rm s}}_{R,q}(n,r)}$ is the $R$-linear span of the elements $x_{{\bf i} {\bf j}}$ for ${\bf i},{\bf j}\in I(n,r)$. The algebra $A^{{\rm s}}_{R,q}(n)$ can be identified with a generalized $FRT$-construction with respect to the subset $N:=\{\beta, \gamma\} \subseteq {\cal E}\otimes {\cal E}$ denoted ${\cal M}_{R}(N)$ in [O2, section 5]. It has been pointed out there that it possesses the structure of a bialgebra where comultiplication and augmentation on the generators $x_{{\bf i} {\bf j}}$ are given by

$$\Delta \mbox{\index{${\Delta (-)}$}} (x_{{\bf i} {\bf j}})=\sum_{{\bf k}\in I(n,r)} x_{{\bf i} {\bf k}} \otimes x_{{\bf k} {\bf j}} ,\; \; \epsilon \mbox{\index{${\epsilon (-) }$}} (x_{{\bf i} {\bf j}}) =\delta_{{\bf i}{\bf j}}.$$ (4)

In particular, the homogeneous summands $A^{{\rm s}}_{R,q}(n,r)$ are subcoalgebras. Furthermore, the tensor space $V^{\otimes r}$ is an $A^{{\rm s}}_{R,q}(n)$ (resp. $A^{{\rm s}}_{R,q}(n,r)$)-(right-)comodule. The structure map $\tau_{r}:V^{\otimes r}\rightarrow V^{\otimes r}\otimes A^{{\rm s}}_{R,q}(n,r)$ is defined by

$$\tau_{r} (v_{{\bf j}})=\sum_{{\bf i}\in I(n,r)} v_{{\bf i}}\otimes x_{{\bf i} {\bf j}}.$$

Now, if $q^2-1$ is an invertible element in $R$, the endomorphism $\gamma$ is known to be in the algebraic span of $\beta$; explicitly one has

$$\gamma=\frac{q^2\beta^{-1}-\beta}{q^2-1}+{\rm id}_{V^{\otimes 2}}.$$

Thus, by [O2, Corollary 2.3] the relations $\gamma\wr x_{{\bf i} {\bf j}}= x_{{\bf i} {\bf j}}\wr \gamma$ are redundant in this case. The reader may check that under these circumstances our bialgebra $A^{{\rm s}}_{R,q}(n)$ is identical to the matrix bialgebra of the usual FRT-construction $F_{R}(n)/\left< \beta\wr X_{{\bf i} {\bf j}}-X_{{\bf i} {\bf j}}\wr \beta,\;\; {\bf i},{\bf j} \in I(n,2)\right>$ connected with the symplectic group for example denoted ${\cal F}_{\beta}(M_n)$ in [CP, 7.3 c].

On the other hand, if $q^2-1$ is not invertible we really need to add the relations $\gamma\wr x_{{\bf i} {\bf j}}= x_{{\bf i} {\bf j}}\wr \gamma$. For instance, it has been proved in [O2, Corollary 6.2] that, setting $q=1$, the bialgebra $A^{{\rm s}}_{R,q}(n)$ is the coordinate ring of the symplectic monoid scheme ${\rm SpM}_{n}(R)$ which is defined by

$${{\rm SpM}_{n}(R)}$$$$\mbox{\index{${{\rm SpM}_{n}(R)}$}}$$$$:= \{ A \in {\rm M}_{n}(R)\vert\;\exists \;\; d(A)\in {R},\; A^tJA=AJA^t=d(A)J\}.$$

Here, $J$ is the Gram-matrix of the canonical skew bilinear form, that is $J=(J_{ij})_{i,j \in \underline{n}}$ where $J_{ij}:=\epsilon_i\delta_{ij'}$. The regular function ${d}$$:{\rm SpM}_{n}(R) \rightarrow {R}$ is called the coefficient of dilation (cf. [Dt]). On the other hand, in this case the bialgebra of the usual FRT-construction equals $A_{R}(n)=R[x_{1 1}, x_{1 2}, \ldots , x_{n n}]$, the commutative polynomial ring in the $x_{i j}$, which is just the coordinate ring of the monoid scheme ${\rm M}_{n}(R)$ of $n \times n$-matrices. Consequently the bialgebra of the usual FRT-construction contains $(q^2-1)$-torsion elements considered over the ground ring $R={\mathbb{Z}}\,[q,q^{-1}]$ of integer Laurent polynomials in $q$.

Let us write down a couple of consequent relations holding in $A^{{\rm s}}_{R,q}(n)$. For this purpose the algebraic span of the $V^{\otimes r}$-endomorphisms

$${\beta_i}$$$$\mbox{\index{${\beta_i}$}}$$$$:= {\rm id}_{V^{\otimes i-1}} \otimes \beta \otimes{\rm id}_{V^{\otimes r-i-1}}\;\;$$ and $$\;\; {\gamma_i}$$$$\mbox{\index{${\gamma_i}$}}$$$$:={\rm id}_{V^{\otimes i-1}} \otimes \gamma \otimes{\rm id}_{V^{\otimes r-i-1}}\;\; i=1, \ldots , r-1$$

in ${\rm End}_{R}{(V^{\otimes r})}$ will be denoted by ${{\cal A}_r}$ (for all $r>1$). According to [O2, section 1, 5] in $A^{{\rm s}}_{R,q}(n,r)$ the following relations hold for all $r>1$:

$$\mu \wr x_{{\bf i} {\bf j}} = x_{{\bf i} {\bf j}} \wr \mu \; \; \mu \in {\cal A}_r, \; {\bf i},{\bf j} \in I(n,r).$$ (5)

The reader should also note that by [O2, Lemma 2.2] all elements of ${\cal A}_r$ must be morphisms of $A^{{\rm s}}_{R,q}(n,r)$-comodules.