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Following the recent trend in explicit neural memory structures, we present anew design of an external memory, wherein memories are stored in an Euclideankey space $\mathbb R^n$. An LSTM controller performs read and write viaspecialized read and write heads. It can move a head by either providing a newaddress in the key space (aka random access) or moving from its previousposition via a Lie group action (aka Lie access). In this way, the "L" and "R"instructions of a traditional Turing Machine are generalized to arbitraryelements of a fixed Lie group action. For this reason, we name this new modelthe Lie Access Neural Turing Machine, or LANTM.  We tested two different configurations of LANTM against an LSTM baseline inseveral basic experiments. We found the right configuration of LANTM tooutperform the baseline in all of our experiments. In particular, we trainedLANTM on addition of $k$-digit numbers for $2 \le k \le 16$, but it was able togeneralize almost perfectly to $17 \le k \le 32$, all with the number ofparameters 2 orders of magnitude below the LSTM baseline.