Download Higher Linear Algebra - University of California, Riverside

Higher Linear Algebra
Alissa S. Crans
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Advised by: John C. Baez
University of California, Riverside
E-mail: [email protected]
June 2004
Higher Dimensional Algebra VI: Lie 2-algebras
Available at: http://arxiv.org/math.QA/0307263
Categorified vector spaces
• Kapranov and Voevodsky defined a
finite-dimensional 2-vector space to be a
category of the form Vectn.
• Instead, we define a 2-vector space to be
a category in Vect.
We have a 2-category, 2Vect whose:
– objects are 2-vector spaces,
– 1-morphisms are linear functors between
these,
– 2-morphisms are linear natural transformations between these
• Proposition: 2Vect ' 2Term
A 2-vector space, V , consists of:
• a vector space V0 of objects,
• a vector space V1 of morphisms,
together with:
• linear source and target maps
s, t : V1 → V0,
• a linear identity-assigning map
i : V0 → V1,
• a linear composition map
◦ : V1 ×V0 V1 → V1
such that the following diagrams commute, expressing the usual
category laws:
• laws specifying the source and target of identity morphisms:
i /
V1
@@
@@
@@ s
V0 @
1 V0
Ã
i /
V
V0 @
1
@@
@@
t
1V0 @@Ã ²
²
V0
V0
• laws specifying the source and target of composite morphisms:
V1 ×V0 V1
◦
p1
V1
s
/
/
V1
p2
s
²
◦
V1 ×V0 V1
V1
/
²
t
²
V0
t
V1
/
²
V0
• the associative law for composition of morphisms:
V1 ×V0 V1 ×V0 V1
◦×V0 1
/
V1 ×V0 V1
1×V0 ◦
◦
²
◦
V1 ×V0 V1
/
²
V1
• the left and right unit laws for composition of morphisms:
V0 ×V0 FV1
i×1 /
V1 ×V0 V1 o 1×i V1 ×V0 V0
FF
FF
FF
FF
F
p2 FFFF
FF
FF
"
xx
xx
x
xx
xx
◦
x
xx p1
xx
x
² |xxx
V1
n-vector spaces
• An (n+1)-vector space is a strict n-category
in Vect.
We have a n-category, nVect whose:
– objects are n-vector spaces,
– 1-morphisms are strict linear functors between these,
– 2-morphisms are strict linear natural transformations between these, . . .
and so on!
• Proposition: (n + 1)Vect ' (n + 1)Term
L N
• ,
of n-vector spaces
Moral: Homological algebra is secretly
categorified linear algebra!
Questions still remain:
• What about weak n-categories in Vect?
• What about weakening laws governing
addition and scalar multiplication?
Categorified Linear Algebra
Associative algebras A∞-algebras
Commutative algebras E∞-algebras
Lie algebras
L∞-algebras
Coalgebras
?
Hopf Algebras
?
..
..
Given a vector space V and an isomorphism
B : V ⊗ V → V ⊗ V,
we say B is a Yang–Baxter operator if it
satisfies the Yang–Baxter equation, which
says that:
(B ⊗1)(1⊗B)(B ⊗1) = (1⊗B)(B ⊗1)(1⊗B),
or in other words, that this diagram commutes:
V ⊗ V ⊗ VUUUU
jj
jjjj
j
1⊗B
j
j
jjj
jjjj
j
j
j
ju
UUUU
UUB⊗1
UUUU
UUUU
UUUU
*
V ⊗V ⊗V
V ⊗V ⊗V
1⊗B
B⊗1
²
²
V ⊗ V ⊗ TVTTT
TTTT
TTTT
TT
B⊗1 TTTTTT)
V ⊗V ⊗V
ii
iiii
i
i
i
iiii
iiiiB⊗1
i
i
i
it i
V ⊗V ⊗V
If we draw B : V ⊗ V → V ⊗ V as a braiding:
V
V
B=
V
V
the Yang–Baxter equation says that:
V
V
V
V
V
V
=
%%
V
%%
%%
%%
%%
%
V
V
V
V
V
Proposition: Let L be a vector space over k
equipped with a skew-symmetric bilinear
operation
[·, ·] : L × L → L.
Let L0 = k ⊕ L and define the isomorphism
B : L0 ⊗ L0 → L0 ⊗ L0 by
B((a, x)⊗(b, y)) = (b, y)⊗(a, x)+(1, 0)⊗(0, [x, y]).
Then B is a solution of the Yang–Baxter
equation if and only if [·, ·] satisfies the Jacobi
identity.
Goal
Develop a higher-dimensional analogue,
obtained by categorifying everything in sight!
A semistrict Lie 2-algebra consists of:
• a 2-vector space L
equipped with:
• a skew-symmetric bilinear functor, the bracket,
[·, ·] : L × L → L
• a completely antisymmetric trilinear natural
isomorphism, the Jacobiator,
Jx,y,z : [[x, y], z] → [x, [y, z]] + [[x, z], y],
that is required to satisfy:
• the Jacobiator identity:
J[w,x],y,z ◦ [Jw,x,z , y] ◦ (Jw,[x,z],y + J[w,z],x,y + Jw,x,[y,z]) =
[Jw,x,y , z] ◦ (J[w,y],x,z + Jw,[x,y],z ) ◦ [Jw,y,z , x] ◦ [w, Jx,y,z ]
for all w, x, y, z ∈ L0.
[[[w, x], y], z]
kkk
kkk
k
k
[Jw,x,y ,z]kkk
k
kkk
kkk
k
k
kk
ku kk
[[[w, y], x], z] + [[w, [x, y]], z]
SSS
SSS
SSS
SSS 1
SSS
SSS
SSS
SSS
S)
[[[w, x], y], z]
J[w,y],x,z +Jw,[x,y],z
J[w,x],y,z
²
[[[w, y], z], x] + [[w, y], [x, z]]
+[w, [[x, y], z]] + [[w, z], [x, y]]
²
[[[w, x], z], y] + [[w, x], [y, z]]
[Jw,x,z ,y]
[Jw,y,z ,x]
²
[[[w, z], y], x] + [[w, [y, z]], x]
+[[w, y], [x, z]] + [w, [[x, y], z]] + [[w, z], [x, y]]
RRR
RRR
RRR
RRR
RRR
R
[w,Jx,y,z ] RRRRR
RRR
RR)
²
[[w, [x, z]], y]
+[[w, x], [y, z]] + [[[w, z], x], y]
lll
lll
l
l
lll Jw,[x,z],y
lll
l
l
l
+J[w,z],x,y + Jw,x,[y,z]
lll
lll
l
l
lu
[[[w, z], y], x] + [[w, z], [x, y]] + [[w, y], [x, z]]
+[w, [[x, z], y]] + [[w, [y, z]], x] + [w, [x, [y, z]]]
Zamolodchikov tetrahedron equation
Given a 2-vector space V and an invertible linear
functor B : V ⊗V → V ⊗V , a linear natural isomorphism
Y : (B ⊗ 1)(1 ⊗ B)(B ⊗ 1) ⇒ (1 ⊗ B)(B ⊗ 1)(1 ⊗ B)
satisfies the Zamolodchikov tetrahedron equation
if:
[Y ◦ (1⊗1⊗B)(1⊗B⊗1)(B⊗1⊗1)][(1⊗B⊗1)(B⊗1⊗1) ◦ Y ◦ (B⊗1⊗1)]
[(1⊗B⊗1)(1⊗1⊗B) ◦ Y ◦ (1⊗1⊗B)][Y ◦ (B⊗1⊗1)(1⊗B⊗1)(1⊗1⊗B)]
=
[(B⊗1⊗1)(1⊗B⊗1)(1⊗1⊗B) ◦ Y ][(B⊗1⊗1) ◦ Y ◦ (B⊗1⊗1)(1⊗B⊗1)]
[(1⊗1⊗B) ◦ Y ◦ (1⊗1⊗B)(1⊗B⊗1)][(1⊗1⊗B)(1⊗B⊗1)(B⊗1⊗1) ◦ Y ]
We should think of Y as the surface in 4-space traced
out by the process of performing the third Reidemeister
move:
Y:
⇒
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%%
%%
%%
%%
%
Left side of Zamolodchikov tetrahedron
equation:
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Right side of Zamolodchikov tetrahedron
equation:
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In short, the Zamolodchikov tetrahedron
equation is a formalization of this commutative
octagon:
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Theorem: Let L be a 2-vector space, let [·, ·] : L×L → L be a skewsymmetric bilinear functor, and let J be a completely antisymmetric
trilinear natural transformation with
Jx,y,z : [[x, y], z] → [x, [y, z]] + [[x, z], y].
Let L0 = K ⊕ L, where K is the categorified ground field.
Let B : L0 ⊗ L0 → L0 ⊗ L0 be defined as follows:
B((a, x) ⊗ (b, y)) = (b, y) ⊗ (a, x) + (1, 0) ⊗ (0, [x, y])
whenever (a, x) and (b, y) are both either objects or morphisms in
L0. Finally, let
Y : (B ⊗ 1)(1 ⊗ B)(B ⊗ 1) ⇒ (1 ⊗ B)(B ⊗ 1)(1 ⊗ B)
be defined as follows:
L0 ⊗ L0 ⊗ L0
p⊗p⊗p
²
L⊗L⊗L
(x,y,z)
J +3
Y =
[[x,y],z]
$
L
z
[x,[y,z]]+[[x,z],y]
aÄ _
j
0
²
L ⊗ L0 ⊗ L0
(1,0)⊗(1,0)⊗(0,a)
where a is either an object or morphism of L. Then Y is a solution
of the Zamolodchikov tetrahedron equation if and only if J satisfies
the Jacobiator identity.
Hierarchy of Higher Commutativity
Topology
Algebra
Crossing
Commutator
Crossing of crossings Jacobi identity
Crossing of crossing
Jacobiator
of crossings
identity
..
..
Examples
Theorem: There is a one-to-one
correspondence between equivalence classes of
Lie 2-algebras (where equivalence is as objects of
the 2-category Lie2Alg) and isomorphism classes
of quadruples consisting of a Lie algebra g, a
vector space V , a representation ρ of g on V ,
and an element of H 3(g, V ).
Example: Given a finite-dimensional, simple
Lie algebra g over C, H 3(g, C) = C, so we can
construct a Lie 2-algebra with nontrivial
Jacobiator consisting of:
• L0 = g
• L1 = identity morphisms
• Jx,y,z = f (x, y, z)1[[x,y],z] where f : g3 → C
is a nontrivial 3-cocycle
In fact, we can take Jx,y,z = ~hx, [y, z]i1 for
~ ∈ C.
Lie 2-groups
A coherent 2-group is a weak monoidal category C in which every morphism is invertible
and every object x ∈ C is equipped with an
adjoint equivalence (x, x̄, ix, ex).
A Lie 2-group is a coherent 2-group object in
DiffCat.