In mathematics , symmetric algebras are used to define polynomials over any vector space . They play an important role, for example, in the theory of Lie groups and in the theory of characteristic classes .
Formal definition Let it be a vector space over a body . Be further
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{\ displaystyle T ^ {k} (V) = \ underbrace {V \ otimes \ cdots \ otimes V} _ {k {\ text {times}}}}
the -fold tensor product of with the conventions and . The direct sum
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{\ displaystyle T ^ {0} (V) = K}
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{\ displaystyle T ^ {1} (V) = V}
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{\ displaystyle T (V) = \ bigoplus _ {k = 0} ^ {\ infty} T ^ {k} (V)}
is the tensor algebra of .
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The two-sided, homogeneous ideal is generated by differences of elementary tensors with "reversed order":
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{\ displaystyle I (V) \ subseteq T (V)}
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{\ displaystyle I (V): = \ mathrm {span} \ left \ {v \ otimes ww \ otimes v {\ Big |} \; v, w \ in V \ right \}}
The symmetric algebra is then defined as the quotient space
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{\ displaystyle S (V) = T (V) / I (V)}
The -th symmetric power of is defined as the image of in , it is denoted by. You have a decomposition
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{\ displaystyle T ^ {k} (V)}
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{\ displaystyle S (V)}
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{\ displaystyle S ^ {k} (V)}
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{\ displaystyle S (V) = \ bigoplus _ {k = 0} ^ {\ infty} S ^ {k} (V)}
The product in symmetric algebra is traditionally written as.
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Similarly, one can define the symmetric algebra of modules using commutative rings .
Examples For is isomorphic to the polynomial ring .
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{\ displaystyle K [X]}
In general, the elements of can be interpreted as polynomials in the elements of a firmly chosen basis of .
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Especially for the vector space of - matrices over , you can see the elements of interpreted as polynomials in the entries of the matrices:
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{\ displaystyle V: = {\ mathfrak {g}} l (n, K) = Mat (n, K)}
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{\ displaystyle S ({\ mathfrak {g}} l (n, K)) \ simeq K \ left [x_ {11}, \ ldots, x_ {nn} \ right]}
Polynomials over vector spaces
Polynomials of degree over a vector space are - by definition - the elements from , where denotes the dual space . These polynomials are linear maps
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{\ displaystyle \ mathbb {K}}
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{\ displaystyle S ^ {k} (V ^ {*})}
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{\ displaystyle P: \ underbrace {V \ otimes \ cdots \ otimes V} _ {k {\ text {-mal}}} \ rightarrow \ mathbb {K}}
which are invariant under the action of the symmetric group . (Note that such a polynomial is already uniquely determined by its values for all of them .)
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{\ displaystyle P (x, x, \ ldots, x)}
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{\ displaystyle x \ in V}
The product
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{\ displaystyle S ^ {k} (V ^ {*}) \ otimes S ^ {l} (V ^ {*}) \ rightarrow S ^ {k + l} (V ^ {*})}
is defined by
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{\ displaystyle (PQ) (v_ {1}, \ ldots, v_ {k + l}) = {\ frac {1} {(k + l)!}} \ sum _ {\ sigma \ in S_ {k + l}} P (v _ {\ sigma (1)}, \ ldots .v _ {\ sigma (k)}) Q (v _ {\ sigma (k + 1)}, \ ldots, v _ {\ sigma (k + l )})}
See also literature
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![K [X]](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b)
![S (\ mathfrak gl (n, K)) \ simeq K \ left [x_ {11}, \ ldots, x_ {nn} \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb126e7a0f926e3433e51991848f8188e8af1b7)
