The degenerate affine braid algebra ${𝔹}_k$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 7 April 2011

The degenerate affine braid algebra ${𝔹}_k$

Let $C$ be a commutative ring. The group algebra of the symmetric group $S_k$ is

$$C{S}_k=C\text{-span}\left\{t_w|w\in S_k\right\},\text{ with }{t}_u{t}_v=t_{uv},$$

(CSk)

for $u,v\in S_k$. Let $s_1,\dots ,s_{k-1}$ be the simple reflections in $S_k$ so that $s_i=\left(i,i+1\right)$.

The degenerate affine braid algebra is the algebra ${𝔹}_k$ over $C$ generated by

$t_u,\text{with}u\in S_k,{\kappa }_0,{\kappa }_1,\text{and}{y}_1,\dots ,y_k,$

(gbgA)

with relations

$t_u{t}_v=t_{uv},y_i{y}_j=y_j{y}_i,{\kappa }_0{\kappa }_1={\kappa }_1{\kappa }_0,{\kappa }_0{y}_i=y_i{\kappa }_0,{\kappa }_1{y}_i=y_i{\kappa }_1,$

(gba1)

${\kappa }_0{t}_{s_i}=t_{s_i}{\kappa }_0,{\kappa }_1{t}_{s_1}{\kappa }_1{t}_{s_1}=t_{s_1}{\kappa }_1{t}_{s_1}{\kappa }_1\text{and}{\kappa }_1{t}_{s_j}=t_{s_j}{\kappa }_1,\text{for }j\ne 1,$

(gba2)

$t_{s_i}\left(y_i+y_{i+1}\right)=\left(y_i+y_{i+1}\right)t_{s_i},\text{and}{y}_j{t}_{s_i}=t_{s_i}{y}_j,\text{and}j\ne i,i+1,$

(gba3)

and

$t_{s_i}{t}_{s_{i+1}}{t}_{i,i+1}{t}_{i+1}{t}_i=t_{i+1,i+2},\text{where}{t}_{i,i+1}=y_{i+1}-t_{s_i}{y}_i{t}_{s_i},$

(gba4)

for $i=1,\dots ,k-1$.

In the degenerate affine braid group ${𝔹}_k$ let $c_0={\kappa }_0$ and

$c_j=\kappa +2(y_1+\dots +y_j)\text{so that}{y}_j=\frac{1}{2}\left(c_j-c_{j-1}\right),\text{for }j=1,\dots ,k,$

(ctoy)

Then the relations (gba3) are equivalent to

$t_{s_i}{c}_j=c_j{t}_{s_i},\text{for}j\ne i.$

(gbp3)

The degenerate affine braid algebra ${𝔹}_k$ has another presentation by generators $C{S}_k$

$t_u,\text{<em>with</em>}u\in S_k,{\kappa }_0,\dots ,{\kappa }_k,\text{<em>and</em>}{t}_{i,j},\text{<em>for</em>}0\le i,j\le k\text{<em>with</em>}i\ne j,$

(gbgB)

and relations

$t_u{t}_v=t_{uv},t_w{\kappa }_i{t}_{w^{-1}}={\kappa }_{w\left(i\right)},t_w{t}_{i,j}{t}_{w^{-1}}=t_{w\left(i\right),w\left(j\right)},$

(gba5)

${\kappa }_i{\kappa }_j={\kappa }_j{\kappa }_i,{\kappa }_i{t}_{\ell ,m}=t_{\ell ,m}{\kappa }_i,$

(gba6)

$t_{i,j}=t_{j,i},t_{p,r}{t}_{\ell ,m}=t_{\ell ,m}{t}_{p,r},t_{i,j}(t_{i,r}+t_{j,r})=(t_{i,r}+t_{j,r})t_{i,j},$

(gba7)

for $p\ne \ell$ and $p\ne m$ and $r\ne \ell$ and $r\ne m$ and $i\ne j$, $i\ne r$ and $j\ne r$.

The commutation relations between the ${\kappa }_i$ and the $t_{i,j}$ can be written in the form

$[{\kappa }_r,t_{\ell ,m}]=0,[t_{i,j},t_{\ell ,m}],\text{and}[t_{i,j},t_{i,m}]=[t_{i,m},t_{j,m}],$

(gba8)

for all $r$ and all $i\ne \ell$ and $i\ne m$ and $j\ne \ell$ and $j\ne m$.

Proof: The generators in (gbgB) are written in terms of the generators in (gbgA) by the formulas

${\kappa }_0={\kappa }_0,{\kappa }_1={\kappa }_1,t_w=t_w,$

(AtoB1)

$t_{0,1}=y_1-\frac{1}{2}{\kappa }_1,\text{and}{t}_{j,j+1}=y_{j+1}-t_{s_j}{y}_j{t}_{s_j},\text{for}j=1,\dots ,k-1,$

(AtoB2)

and

${\kappa }_m=t_u{\kappa }_1{t}_{u^{-1}},t_{0,m}=t_u{t}_{0,1}{t}_{u^{-1}}\text{and}{t}_{i,j}=t_v{t}_{1,2}{t}_{v^{-1}},$

(AtoB3)

for $u,v\in S_k$ such that $u\left(1\right)=m$, $v\left(1\right)=i$ and $v\left(2\right)=j$.

The generators in (gbgA) are written in terms of the generators in (gbgB) by the formulas

${\kappa }_0={\kappa }_0,{\kappa }_1={\kappa }_1,t_w=t_w,\text{and}{y}_j=\frac{1}{2}{\kappa }_j+\sum _{0\le \ell <j}{t}_{\ell ,j}.$

(BtoA)

By the first formula in (ctoy) and the last formula in (BtoA)

$$c_j=\sum _{i=0}^j{\kappa }_i+2\sum _{0\le 𝓁<m\le j}{t}_{𝓁,m}.$$

(cytokt)

		Let us show that the relations in (gba1-gba4) follow from the relations in (gba5-gba7).
	(a)   The relation $t_u{t}_v=t_{uv}$ in (gba1) is the first relation in (gba5). (b)   The relation $y_i{y}_j=y_j{y}_i$ in (gba1): Assume that $i<j$. Using the relations in (gba6) and (gba7), $\begin{array}{rl}[y_i,y_j]& =[\frac{1}{2}{\kappa }_i+\sum _{\ell <i}{t}_{\ell ,i},\frac{1}{2}{\kappa }_j+\sum _{m<j}{t}_{m,j}]=[\sum _{\ell <i}{t}_{\ell ,i},\sum _{m<j}{t}_{m,j}]\\ & =\sum _{\ell <i}[t_{\ell ,i},\sum _{m<j}{t}_{m,j}]=\sum _{\ell <i}[t_{\ell ,i},(t_{\ell ,j}+t_{i,j})+\sum _{\begin{array}{c}m<j\\ m\ne \ell ,m\ne j\end{array}}{t}_{m,j}]=0.\end{array}$ (c)   The relation ${\kappa }_0{\kappa }_1={\kappa }_1{\kappa }_0$ in (gba1) is part of the first relation in (gba6), and the relations ${\kappa }_0{y}_i=y_i{\kappa }_0$ and ${\kappa }_1{y}_i=y_i{\kappa }_1$ in (gba1) follow from the relations ${\kappa }_i{\kappa }_j={\kappa }_j{\kappa }_i$ and ${\kappa }_i{t}_{\ell ,m}=t_{\ell ,m}{\kappa }_i,$ in (gba6). (d)   The relations ${\kappa }_0{t}_{s_i}=t_{s_i}{\kappa }_0$ and ${\kappa }_1{t}_{s_j}=t_{s_j}{\kappa }_1$ for $j\ne 1$ from (gba2) follow from the relation $t_w{\kappa }_i{t}_{w^{-1}}={\kappa }_{w\left(i\right)}$ in (gba5), and the relation ${\kappa }_1{t}_{s_1}{\kappa }_1{t}_{s_1}=t_{s_1}{\kappa }_1{t}_{s_1}{\kappa }_1$ from (gba2) follows from ${\kappa }_1{\kappa }_2={\kappa }_2{\kappa }_1$, which is part of the first relation in (gba6). (e)   The relations in (gba3) and (gba4) all follow from the relations $t_w{\kappa }_i{t}_{w^{-1}}={\kappa }_{w\left(i\right)}$ and $t_w{t}_{i,j}{t}_{w^{-1}}=t_{w\left(i\right),w\left(j\right)}$ in (gba5).

		To complete the proof let us show that the relations in (gba5-gba7) follow from the relations in (gba1-gba4).
	(a)   The relation $t_u{t}_v=t_{uv}$ in (gba5) is the first relation in (gba1). (b)   The relations $t_w{\kappa }_i{t}_{w^{-1}}={\kappa }_{w\left(i\right)}$ in (gba5) follow from the first and last relations in (gba2) (and force the definition of ${\kappa }_m$ in (AtoB3)). (c)   Since $t_{0,1}=y_1-\frac{1}{2}{\kappa }_1$, the relations in (gba6) follow from the last relation in each of (gba2) and (gba3) (and force the definition of $t_{0,m}$ in (AtoB3)). (d)   Since $t_{1,2}=y_2-t_{s_1}{y}_1{t}_{s_1}$, the first relation in (gba3) gives $t_{s_1}{t}_{1,2}{t}_{s_1}-t_{1,2}=(t_{s_1}{y}_2{t}_{s_1}-y_1)-y_2+t_{s_1}{y}_1{t}_{s_1}=t_{s_1}(y_1+y_2)t_{s_1}-(y_1+y_2)=0$. (sts) The relations $t_w{t}_{1,2}{t}_{w^{-1}}=t_{w\left(1\right),w\left(2\right)}$ in (gba5) then follow from (sts) and the last relation in (gba3) (and force the definitions $t_{i,j}=t_v{t}_{1,2}{t}_{v^{-1}}$ in (AtoB3)). (e)   The third relation in (gba1) is ${\kappa }_0{\kappa }_1={\kappa }_1{\kappa }_0$ and the second relation in (gba2) gives ${\kappa }_1{\kappa }_2={\kappa }_2{\kappa }_1$. The relations ${\kappa }_i{\kappa }_j={\kappa }_j{\kappa }_i$ in (gba6) then follow from the second set of relations in (gba5). (f)   The second relation in (gba2) gives $[{\kappa }_1,{\kappa }_2]=0$. Using this and the relations in (gba1), $[{\kappa }_1,t_{0,2}+t_{1,2}]=[{\kappa }_1,(y_2-\frac{1}{2}{\kappa }_2-t_{1,2})+t_{1,2}]=[{\kappa }_1,\frac{1}{2}{\kappa }_2]=0$, (ktc) and $[t_{0,1},t_{0,2}+t_{1,2}]=[y_1-\frac{1}{2}{\kappa }_1,y_2-\frac{1}{2}{\kappa }_2]=\frac{1}{4}[{\kappa }_1,{\kappa }_2]=0$, (ttc) so that $[t_{0,1},{\kappa }_2]=[t_{0,1},2y_2-2(t_{0,2}+t_{1,2}\left)\right]=[t_{0,1},2y_2]=[y_1-\frac{1}{2}{\kappa }_1,2y_2]=-[{\kappa }_1,y_2]=0$. Conjugating the last relation by $t_{s_1}$ gives $[{\kappa }_1,t_{0,2}]=0$,     and thus     $[{\kappa }_1,t_{1,2}]=0$, by (ktc). By the third and fourth relations in (gba1), $[{\kappa }_0,t_{0,1}]=[{\kappa }_0,y_1-\frac{1}{2}{\kappa }_1]=0$,     and     $[{\kappa }_1,t_{0,1}]=[{\kappa }_1,y_1-\frac{1}{2}{\kappa }_1]=0$. By the relations in (gba2) and (gba1), $[{\kappa }_0,t_{1,2}]=[{\kappa }_0,y_2-t_{s_1}{y}_1{t}_{s_1}]=0$     and     $[{\kappa }_1,t_{2,3}]=[{\kappa }_1,y_3-t_{s_2}{y}_2{t}_{s_2}]=0$. Putting these together with the (already established) relations in (gba5) provides the second set of relations in (gba6). (g)   From the commutativity of the $y_i$ and the second relation in (gba3) $t_{1,2}{t}_{3,4}=(y_2-t_{s_1}{y}_1{t}_{s_1})(y_4-t_{s_3}{y}_3{t}_{s_3})=(y_4-t_{s_3}{y}_3{t}_{s_3})(y_2-t_{s_1}{y}_1{t}_{s_1})=t_{3,4}{t}_{1,2}$. By the last relation in (gba1) and the last relation in (gba2), $[t_{0,1},t_{2,3}]=[y_1-\frac{1}{2}{\kappa }_1,y_3-t_{s_2}{y}_2{t}_{s_2}]=0$. Together with the (already established) relations in (gba5), we obtain the first set of relations in (gba7). (h)   Conjugating (ttc) by $t_{s_2}{t}_{s_1}{t}_{s_2}$ gives $[t_{0,2},t_{0,3}+t_{2,3}]=0$ and this and the (already established) relations in (gba6) and the first set of relations in (gba7) provide $\begin{array}{rl}0& =[y_2,y_3]=[\frac{1}{2}{\kappa }_2+t_{0,2}+t_{1,2},\frac{1}{2}{\kappa }_3+t_{0,3}+t_{1,3}+t_{2,3}]\\ & =[t_{0,2}+t_{1,2},t_{0,3}+t_{1,3}+t_{2,3}]=[t_{1,2},t_{0,3}+t_{1,3}+t_{2,3}]=[t_{1,2},t_{1,3}+t_{2,3}].\end{array}$ Note also that $\begin{array}{rl}[t_{1,2},t_{1,0}+t_{2,0}]& =[t_{1,2},t_{0,1}+t_{0,2}]=-[t_{0,1},t_{1,2}]+[t_{1,2},t_{0,2}]\\ & =[t_{0,1},t_{0,2}]+[t_{1,2},t_{0,2}]=t_{s_1}[t_{0,2}+t_{1,2},t_{0,1}]t_{s_1}=0,\end{array}$ by (two applications of) (ttc). The last set of relations in (gba7) now follow from the last set of relations in (gba5).

$\square$

Notes and References

To our knowledge, the definition of the degenerate affine braid algebra has not appeared previously in the literature. It appears in [Da] and the presentation above is a part of the forthcoming paper [DRV]. Both the affine Hecke algebra and the affine BMW algebra are quotients of the group algebra of the affine group. In a similar manner it seemed desirable to define a degenerate affine braid algebra which has both the degenerate affine Hecke algebra and the degenerate affine BMW algebras as quotients. This is also justified by the analogy between the corresponding Schur-Weyl duality statements, see Actions of Tantalizers.

Bibliography

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

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