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A f f i n e L i e a l g e b r a s a s c e n t r a l e x t e n s i o n o f l o o p a l g e b r a s
T o g g l e A f f i n e L i e a l g e b r a s a s c e n t r a l e x t e n s i o n o f l o o p a l g e b r a s s u b s e c t i o n
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A f f i n e L i e a l g e b r a
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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Type of Lie algebra of interest in physics
In mathematics , loop algebras are certain types of Lie algebras , of particular interest in theoretical physics .
Definition [ edit ]
For a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
over a field
K
{\displaystyle K}
, if
K
[
t
,
t
−
1
]
{\displaystyle K[t,t^{-1}]}
is the space of Laurent polynomials , then
L
g
:=
g
⊗
K
[
t
,
t
−
1
]
,
{\displaystyle L{\mathfrak {g}}:={\mathfrak {g}}\otimes K[t,t^{-1}],}
with the inherited bracket
[
X
⊗
t
m
,
Y
⊗
t
n
]
=
[
X
,
Y
]
⊗
t
m
+
n
.
{\displaystyle [X\otimes t^{m},Y\otimes t^{n}]=[X,Y]\otimes t^{m+n}.}
Geometric definition [ edit ]
If
g
{\displaystyle {\mathfrak {g}}}
is a Lie algebra, the tensor product of
g
{\displaystyle {\mathfrak {g}}}
with C ∞ (S 1 ) , the algebra of (complex) smooth functions over the circle manifold S 1 (equivalently, smooth complex-valued periodic functions of a given period),
g
⊗
C
∞
(
S
1
)
,
{\displaystyle {\mathfrak {g}}\otimes C^{\infty }(S^{1}),}
is an infinite-dimensional Lie algebra with the Lie bracket given by
[
g
1
⊗
f
1
,
g
2
⊗
f
2
]
=
[
g
1
,
g
2
]
⊗
f
1
f
2
.
{\displaystyle [g_{1}\otimes f_{1},g_{2}\otimes f_{2}]=[g_{1},g_{2}]\otimes f_{1}f_{2}.}
Here g 1 and g 2 are elements of
g
{\displaystyle {\mathfrak {g}}}
and f 1 and f 2 are elements of C ∞ (S 1 ) .
This isn't precisely what would correspond to the direct product of infinitely many copies of
g
{\displaystyle {\mathfrak {g}}}
, one for each point in S 1 , because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from S 1 to
g
{\displaystyle {\mathfrak {g}}}
; a smooth parametrized loop in
g
{\displaystyle {\mathfrak {g}}}
, in other words. This is why it is called the loop algebra .
Gradation [ edit ]
Defining
g
i
{\displaystyle {\mathfrak {g}}_{i}}
to be the linear subspace
g
i
=
g
⊗
t
i
<
L
g
,
{\displaystyle {\mathfrak {g}}_{i}={\mathfrak {g}}\otimes t^{i}<L{\mathfrak {g}},}
the bracket restricts to a product
[
⋅
,
⋅
]
:
g
i
×
g
j
→
g
i
+
j
,
{\displaystyle [\cdot \,,\,\cdot ]:{\mathfrak {g}}_{i}\times {\mathfrak {g}}_{j}\rightarrow {\mathfrak {g}}_{i+j},}
hence giving the loop algebra a
Z
{\displaystyle \mathbb {Z} }
-graded Lie algebra structure.
In particular, the bracket restricts to the 'zero-mode' subalgebra
g
0
≅
g
{\displaystyle {\mathfrak {g}}_{0}\cong {\mathfrak {g}}}
.
Derivation [ edit ]
There is a natural derivation on the loop algebra, conventionally denoted
d
{\displaystyle d}
acting as
d
:
L
g
→
L
g
{\displaystyle d:L{\mathfrak {g}}\rightarrow L{\mathfrak {g}}}
d
(
X
⊗
t
n
)
=
n
X
⊗
t
n
{\displaystyle d(X\otimes t^{n})=nX\otimes t^{n}}
and so can be thought of formally as
d
=
t
d
d
t
{\displaystyle d=t{\frac {d}{dt}}}
.
It is required to define affine Lie algebras , which are used in physics, particularly conformal field theory .
Loop group [ edit ]
Similarly, a set of all smooth maps from S 1 to a Lie group G forms an infinite-dimensional Lie group (Lie group in the sense we can define functional derivatives over it) called the loop group . The Lie algebra of a loop group is the corresponding loop algebra.
Affine Lie algebras as central extension of loop algebras [ edit ]
If
g
{\displaystyle {\mathfrak {g}}}
is a semisimple Lie algebra , then a nontrivial central extension of its loop algebra
L
g
{\displaystyle L{\mathfrak {g}}}
gives rise to an affine Lie algebra . Furthermore this central extension is unique.[1]
The central extension is given by adjoining a central element
k
^
{\displaystyle {\hat {k}}}
, that is, for all
X
⊗
t
n
∈
L
g
{\displaystyle X\otimes t^{n}\in L{\mathfrak {g}}}
,
[
k
^
,
X
⊗
t
n
]
=
0
,
{\displaystyle [{\hat {k}},X\otimes t^{n}]=0,}
and modifying the bracket on the loop algebra to
[
X
⊗
t
m
,
Y
⊗
t
n
]
=
[
X
,
Y
]
⊗
t
m
+
n
+
m
B
(
X
,
Y
)
δ
m
+
n
,
0
k
^
,
{\displaystyle [X\otimes t^{m},Y\otimes t^{n}]=[X,Y]\otimes t^{m+n}+mB(X,Y)\delta _{m+n,0}{\hat {k}},}
where
B
(
⋅
,
⋅
)
{\displaystyle B(\cdot ,\cdot )}
is the Killing form .
The central extension is, as a vector space,
L
g
⊕
C
k
^
{\displaystyle L{\mathfrak {g}}\oplus \mathbb {C} {\hat {k}}}
(in its usual definition, as more generally,
C
{\displaystyle \mathbb {C} }
can be taken to be an arbitrary field).
Cocycle [ edit ]
Using the language of Lie algebra cohomology , the central extension can be described using a 2-cocycle on the loop algebra. This is the map
φ
:
L
g
×
L
g
→
C
{\displaystyle \varphi :L{\mathfrak {g}}\times L{\mathfrak {g}}\rightarrow \mathbb {C} }
satisfying
φ
(
X
⊗
t
m
,
Y
⊗
t
n
)
=
m
B
(
X
,
Y
)
δ
m
+
n
,
0
.
{\displaystyle \varphi (X\otimes t^{m},Y\otimes t^{n})=mB(X,Y)\delta _{m+n,0}.}
Then the extra term added to the bracket is
φ
(
X
⊗
t
m
,
Y
⊗
t
n
)
k
^
.
{\displaystyle \varphi (X\otimes t^{m},Y\otimes t^{n}){\hat {k}}.}
Affine Lie algebra [ edit ]
In physics, the central extension
L
g
⊕
C
k
^
{\displaystyle L{\mathfrak {g}}\oplus \mathbb {C} {\hat {k}}}
is sometimes referred to as the affine Lie algebra. In mathematics, this is insufficient, and the full affine Lie algebra is the vector space[2]
g
^
=
L
g
⊕
C
k
^
⊕
C
d
{\displaystyle {\hat {\mathfrak {g}}}=L{\mathfrak {g}}\oplus \mathbb {C} {\hat {k}}\oplus \mathbb {C} d}
where
d
{\displaystyle d}
is the derivation defined above.
On this space, the Killing form can be extended to a non-degenerate form, and so allows a root system analysis of the affine Lie algebra.
References [ edit ]
^ P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory , 1997, ISBN 0-387-94785-X
Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups , Cambridge University Press, ISBN 0-521-48412-X
t
e
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Loop_algebra&oldid=1223586817 "
C a t e g o r i e s :
● L i e a l g e b r a s
● A l g e b r a s t u b s
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n m a t c h e s W i k i d a t a
● A l l s t u b a r t i c l e s
● T h i s p a g e w a s l a s t e d i t e d o n 1 3 M a y 2 0 2 4 , a t 0 3 : 1 9 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w