FRT-Construction

The $r$ fold tensor product of $V$ is denoted by $V^{\otimes r}$. Since it is a free $R$-module, as well, we may apply all results of former sections to a situation where $V$ is substituted by $V^{\otimes r}$, ${\cal E}$ by ${\cal E}_r:={\rm End}_{R}{(V^{\otimes r})}$ and ${{\cal E}}^*$ by ${{\cal E}_r}^*:={\rm Hom}_{R}{({\cal E}_r,R)}$. There are natural isomorphisms between ${\cal E}_r$ and ${{\cal E}}^{\otimes r}$ and between ${{\cal E}_r}^*$ and ${{{\cal E}}^*}^{\otimes r}$. We will omit them in our notation and consider them as identity maps. Set $V^{\otimes 0}:={\cal E}_{0}:=R$.

Suppose we are given a family ${\cal A}=({\cal A}_r)_{r \in {\mathbb{N}}_0}$ of $R$-algebras such that ${\cal A}_r\subseteq {\cal E}_r$. According to lemma 2.1 there is an associated family of coideals $K_r$ in ${{\cal E}_r}^*$.

$$K_r:=K({\cal A}_r).$$ (9)

By the above identification we consider ${\cal T}({{\cal E}}^*):=\bigoplus_{r \in {\mathbb{N}}_0}{{\cal E}_r}^*$ as the tensor algebra on the $R$-module ${{\cal E}}^*$. There is a unique coalgebra structure on this tensor algebra, extending the one from ${{\cal E}}^*$ and turning ${\cal T}({{\cal E}}^*)$ into a bialgebra. Furthermore the ${{\cal E}_r}^*$ are subcoalgebras dual to the algebras ${\cal E}_r$, i.e. the coalgebra structure on ${{\cal E}_r}^*$ is the one considered above. Epimorphic images of this bialgebra are called matric bialgebras (cf. [Ta]). Let us investigate under what circumstances the coideal

$$I:=\oplus_{r \in {\mathbb{N}}_0 } K_r$$ (10)

becomes an ideal in ${\cal T}({{\cal E}}^*)$ and thus consequently is a biideal. To this claim we consider inclusion maps $s_{r},t_{r}:{\cal E}_r\rightarrow {\cal E}_{r+1}$ given by

$$s_{r}(\mu )=\mu \otimes {\rm id}_{V}, \; t_{r}(\mu )={\rm id}_{V}\otimes \mu ,\; \mu \in {\cal E}_r$$

and focus attention to a special situation, which is general enough for our applications. We start with an arbitrary subset $N\subseteq {\cal E}_{2}$ and define a family ${\cal A}$ inductively beginning with ${\cal A}_{0}:=R,\; {\cal A}_{1}:=R\cdot{\rm id}_{V}$ and ${\cal A}_{2}$ as the algebraic span of $N$ in ${\cal E}_{2}$, than continuing by the formula

$${\cal A}_r:=\left<s_{r-1}({\cal A}_{r-1}) + t_{r-1}({\cal A}_{r-1})\right>_{\rm Alg} \;\;\mbox{ for } \;\;r>2.$$

We call this a by $N$ induced family of subalgebras of ${\cal E}_r$.

Proposition 5.1   Let ${\cal A}$ be the family induced by a subset $N\subseteq {\cal E}_{2}$. Then the coideal $I$ defined in (10) is a homogenous biideal in ${\cal T}({{\cal E}}^*)$ generated by the coideal $K_2$.

PROOF: First we show that $I$ is a homogenous ideal. To this claim take $a \in K_r$ and $b \in{{\cal E}_{u}}^*$. We have to show $a \otimes b \in K_{r+u}$ and $b \otimes a \in K_{r+u}$. The choice of the element $a$ can be reduced to a generator $a={\vartheta}_{tr}([\mu,\nu])$ of the $R$-module $K_r$ with $\mu \in{\cal E}_r$ and $\nu \in {\cal A}_r$. Letting $\hat{\nu}:=s_{r+u-1} \circ s_{r+u-2} \circ \ldots \circ s_{r}(\nu) = \nu \otimes {\rm id}_{V^{\otimes u}} \in {\cal A}_{r+u}$ and $\hat\mu :=\mu \otimes \bar b$, where $\bar b\in {\cal E}_{u}$ is the preimage of $b$ under ${\vartheta}_{tr}$ one obtains

$$a \otimes b = {\vartheta}_{tr}([\mu,\nu]) \otimes b = {\vartheta}_{tr}([\hat\mu ,\hat{\nu}]) \in K_{r+u}.$$

Similary $b \otimes a \in K_{r+u}$ and the first statement is established.

For the second part denote by $J$ the homogenous ideal in ${\cal T}({{\cal E}}^*)$ generated by $K_2$. Since $I$ is a homogenous ideal containing $K_2$ we get $J\subseteq I$. For the reverse inclusion we show $K_r\subseteq J$ by induction on $r$. Clearly $K_0=K_1=(0)$ and $K_2$ are contained in $J$. Now suppose $r>2$. Letting $J_r:=J\cap{{\cal E}_r}^*$ one obtains $J_r=J_{r-1}\otimes {{\cal E}}^*+{{\cal E}}^*\otimes J_{r-1}$. By induction hypothesis we see $K_{r-1} = J_{r-1}$. Therefore

$$J_r=K_{r-1}\otimes {{\cal E}}^*+ {{\cal E}}^*\otimes K_{r-1}= K(M)$$

where $M:=s_{r-1}({\cal A}_{r-1}) + t_{r-1}({\cal A}_{r-1}) \subseteq {\cal E}_r$. But since ${\cal A}_r$ is generated by $M$ as an algebra we finally see from corollary 2.3 $K_r =K({\cal A}_r)=K(M)=J_r$.$\Box$

According to the proposition we may assign a graded matric bialgebra to each subset $N\subseteq {\cal E}_{2}$ by

$${\cal M}(N):={\cal T}({{\cal E}}^*)/I$$ (11)

whose homogenous summand are the centralizer coalgebras $M({\cal A}_r)$. This bialgebra will be called the FRT-construction corresponding to the subset $N$. The reader familiar with the ordinary FRT-construction will recognize the latter one as the special case where $N$ consists of just one element $\beta$ (use the description (12) below). In this case we write ${\cal M}(\beta):={\cal M}(N)$. Usually in the applications this $\beta$ is a quantum Yang-Baxter operator leading to a representation of the Artin braid groups on the modules $V^{\otimes r}$. In this situaton the algebras ${\cal A}_r$ are just the images of the corresponding group algebras (over $R$) under this representation.

An application where $N$ must consist of two elements will be given in the next section. Here the ${\cal A}_r$ are images of the Brauer centralizer algebras under Brauer's representations corresponding to the symplectic groups. The bialgebra ${\cal M}(N)$ will turn out to be the coordinate ring of a certain symplectic monoid. It is a remarkable fact, that the second operator is only needed in the classical situation, whereas in the quantum case it lies in the algebraic span of the quantum Yang-Baxter operator (see [Oe], Bemerkung 2.5.1, 2.5.4). Thus, the ordinary FRT-construction behaves singularly (in a certain sense) when specializing the deformation parameter to 1, i.e. in the classical limit.

We close this section giving a more convenient description of ${\cal M}(N)$. We denote by $I(n,r)$ the set of maps from from $\underline{r}:=\{1, \ldots , r\}$ to $\underline{n}:=\{1, \ldots , n\}$ and call the elements ${\bf i} \in I(n,r)$ multi-indices writing ${\bf i}=(i_1, \ldots , i_r)$, where $i_j\in \underline{n}$ for $j\in \underline{r}$. The residue classes of the multiplicative generators ${{e}^*_{i}}^{j} +I$ of ${\cal M}(N)$ where $i,j\in \underline{n}$ will be denoted by $x_{i j}$. For pairs of multi-indices ${\bf i},{\bf j}\in I(n,r)$ we introduce the abbreviation

$$x_{{\bf i} {\bf j}}:=x_{i_1 j_1}x_{i_2 j_2}\ldots x_{i_r j_r}.$$

Using the notation introduced in (1) we obtain a presentation of ${\cal M}(N)$ by generators and relations given as follows

$${\cal M}(N)=\left< x_{i j}, \; i,j\in \underline{n}\vert\;\;... ...bf j}}\mu,\; {\bf i}, {\bf j} \in I(n,2),\; \mu\in N\right>.$$ (12)

The verification of this formula follows from the second statement of proposition 5.1 together with (2).