A categorification of the Burau representation at prime roots of unity

Abstract

We construct a p-DG structure on an algebra Koszul dual to a zigzag algebra used by Khovanov and Seidel to construct a categorical braid group action. We show there is a braid group action in this p-DG setting.

Appendix: Maps between simples

For $\lambda =1$ the unique non-trivial map (up to homotopy) $\psi _{i+1} :L_{i+1}[1] \lbrace 2p-1 \rbrace \rightarrow L_i$ is easy to construct. The map between their cofibrant resolutions

and

respectively is given by the identity map $P_{i+1} \{ -2p+3+2r \} \rightarrow P_{i+1} \{-2p+3+2r \}$ for $r=0,\ldots ,p-2$ and is zero otherwise.

For $\lambda =1$ the unique (up to homotopy) map $\phi _i :L_i \rightarrow L_{i+1}[1] \{ 1 \}$ is more difficult to construct. For $p=2$ the map is given by:

(5.23)

For a prime $p>2$ there is a map $\phi _i :L_i \rightarrow L_{i+1}[1] \{ 1 \}$ between the cofibrant resolution of $L_i$

and the cofibrant resolution of $L_{i+1}[1] \{ 1 \}$

given on components as follows:

• $d_j :P_i \rightarrow P_{i+1} \{-2p+3+2j \}$ for $j=0,\ldots ,p-2$

  $d_j = \cdot (-1)^{j+1} (p-j-2)!(i|i+1)(i+1|i|i+1)^{p-j-2}$

• $b :P_i \rightarrow P_{i+2} \{ -2p+2 \}$

  $b = \cdot -(i|i+1|i+2)(i+2|i+1| i+2)^{p-2}$

• $p_{k,j} :P_{i+1} \{ -1-2k \} \rightarrow P_{i+1} \{ -2p+3+2j \}$ for $0 \le j \le p-3-k$

  $p_{k,j} = \cdot (-1)^{k+j} \frac{(p-2-j)!}{(k+1)!} (i+1|i|i+1)^{p-j-k-2}$

• $p_{k,p-2-k} :P_{i+1} \{ -1-2k \} \rightarrow P_{i+1} \{ -2k-1 \}$ for $0 \le k \le p-2$

  $p_{k,p-2-k} = -1$

• $h_{k,j} :P_{i-1} \{-1-2k \} \rightarrow P_{i+1} \{ -2p+3+2j \}$ for $0 \le j \le p-3-k$

  $h_{k,j} = \cdot \frac{(-1)^{k+1} k}{(k+1)!(j+1)!} (i-1|i|i+1)(i+1|i|i+1)^{p-j-k-3}$

• $m_k :P_{i+1} \{-1-2k \} \rightarrow P_{i+2} \{-2p+2 \}$ for $0 \le k \le p-3$

  $m_k=\cdot \frac{(-1)^k}{(k+1)!} (i+1|i+2)(i+2|i+1|i+2)^{p-k-2}$

• $m_{p-2} :P_{i+1} \{-2p+3 \} \rightarrow P_{i+2} \{-2p+2 \}$

  $m_{p-2} = -(i+1|i+2)$

• $n_{p-2} :P_{i+1} \{-2p+3 \} \rightarrow P_{i} \{-2p+2 \}$

  $n_{p-2} = 2(i+1|i)$

• $f_k :P_{i-1} \{ -1-2k \} \rightarrow P_{i+2} \{-2p+2 \}$ for $1 \le k \le p-3$

  $f_k = \cdot \frac{(-1)^{k+1} k}{(k+1)!} (i-1|i|i+1|i+2)(i+2|i+1|i+2)^{p-k-3}$

• $g_{p-2} :P_{i-1} \{-2p+3\} \rightarrow P_i \{-2p+2\}$

  $g_{p-2}= 2(i-1|i)$

• $\gamma :P_i \{-2p+2 \} \rightarrow P_i \{-2p+2 \}$

  $\gamma =1$.