  Session-based recommendation (SBR) learns users' preferences by capturing the
short-term and sequential patterns from the evolution of user behaviors. Among
the studies in the SBR field, graph-based approaches are a relatively powerful
kind of way, which generally extract item information by message aggregation
under Euclidean space. However, such methods can't effectively extract the
hierarchical information contained among consecutive items in a session, which
is critical to represent users' preferences. In this paper, we present a
hyperbolic contrastive graph recommender (HCGR), a principled session-based
recommendation framework involving Lorentz hyperbolic space to adequately
capture the coherence and hierarchical representations of the items. Within
this framework, we design a novel adaptive hyperbolic attention computation to
aggregate the graph message of each user's preference in a session-based
behavior sequence. In addition, contrastive learning is leveraged to optimize
the item representation by considering the geodesic distance between positive
and negative samples in hyperbolic space. Extensive experiments on four
real-world datasets demonstrate that HCGR consistently outperforms
state-of-the-art baselines by 0.43$\%$-28.84$\%$ in terms of $HitRate$, $NDCG$
and $MRR$.
