  We present results from a new ultra-deep 400 ks Chandra observation of the
SSA22 protocluster at z = 3.09. We have studied the X-ray properties of 234 z ~
3 Lyman break galaxies (LBGs; protocluster and field) and 158 z = 3.09 Ly-alpha
emitters (LAEs) in SSA22 to measure the influence of the high-density
protocluster environment on the accretion activity of supermassive black holes
(SMBHs) in these UV-selected star forming populations. We detect individually
X-ray emission from active galactic nuclei (AGNs) in six LBGs and five LAEs;
due to small overlap between the LBG and LAE source population, ten of these
sources are unique. At least six and potentially eight of these sources are
members of the protocluster. These sources have rest-frame 8-32 keV
luminosities in the range of L_8-32 keV = (3-50) X 10^{43} ergs/s and an
average observed-frame 2-8 keV to 0.5-2 keV band-ratio of ~0.8 (mean effective
photon index of Gamma_eff = 1.1), suggesting significant absorption columns of
N_H > 10^{22}-10^{24} cm^{-2}. We find that the fraction of LBGs and LAEs in
the z = 3.09 protocluster harboring an AGN with L_8-32 keV > 3 X 10^{43} ergs/s
is 9.5^{+12.7}_{-6.1}% and 5.1^{+6.8}_{-3.3}%, respectively. These AGN
fractions are somewhat larger (by a mean factor of 6.1^{+10.3}_{-3.6};
significant at the 95% confidence level) than z ~ 3 sources found in
lower-density "field" environments. Theoretical models imply that these results
may be due to the presence of more actively growing and/or massive SMBHs in
LBGs and LAEs within the protocluster compared to the field. Such a result is
expected in a scenario where enhanced merger activity in the protocluster
drives accelerated galaxy and SMBH growth at z > 2-3. (abridged)
