Euclidean group

In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is the free commutative unital associative algebra over K containing V.

It corresponds to polynomials with indeterminates in V, without choosing coordinates. The dual, S(V*) corresponds to polynomials on V.

It should not be confused with symmetric tensors in V. A Frobenius algebra whose bilinear form is symmetric is also called a symmetric algebra, but is not discussed here.

Construction

It turns out that S(V) is in effect the same as the polynomial ring, over K, in indeterminates that are basis vectors for V. Therefore this construction only brings something extra when the "naturality" of looking at polynomials this way has some advantage.

It is possible to use the tensor algebra T(V) to describe the symmetric algebra S(V). In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative; if elements of V commute, then tensors in them must, so that we construct the symmetric algebra from the tensor algebra by taking the quotient algebra of T(V) by the ideal generated by all differences of products

$v\otimes w-w\otimes v.$

for v and w in V.

Grading

Just as with a polynomial ring, there is a direct sum decomposition of S(V) as a graded algebra, into summands

S^k(V)

which consist of the linear span of the monomials in vectors of V of degree k, for k = 0, 1, 2, ... (with S⁰(V) = K and S¹(V)=V). The K-vector space S^k(V) is the k-th symmetric power of V. (The case k = 2, for example, is the symmetric square and denoted Sym²(V).) It has a universal property with respect to symmetric multilinear operators defined on V^k.

In terms of the tensor algebra grading, S^k(V) is the quotient space of T^k(V) by the subspace generated by all differences of products

$v\otimes w-w\otimes v.$

and products of these with other algebra elements.

Distinction with symmetric tensors

The symmetric algebra and symmetric tensors are easily confused: the symmetric algebra is a quotient of the tensor algebra, while the symmetric tensors are a subspace of the tensor algebra.

The symmetric algebra must be a quotient to satisfy its universal property (since every symmetric algebra is an algebra, the tensor algebra maps to the symmetric algebra).

Conversely, symmetric tensors are defined as invariants: given the natural action of the symmetric group on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially. Note that under the tensor product, symmetric tensors are not a subalgebra: given vectors v and w, they are trivially symmetric 1-tensors, but v ⊗ w is not a symmetric 2-tensor.

The grade 2 part of this distinction is the difference between symmetric bilinear forms (symmetric 2-tensors) and quadratic forms (elements of the symmetric square), as described in ε-quadratic forms.

In characteristic 0 symmetric tensors and the symmetric algebra can be identified. In any characteristic, there is a symmetrization map from the symmetric algebra to the symmetric tensors, given by:

$v_1\cdots v_k\mapsto \sum _{\sigma \in S_k}{v}_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (k)}.$

The composition with the inclusion of the symmetric tensors in the tensor algebra and the quotient to the symmetric algebra is multiplication by k! on the kth graded component.

Thus in characteristic 0, the symmetrization map is an isomorphism of graded vector spaces, and one can identify symmetric tensors with elements of the symmetric algebra. One divides by k! to make this a section of the quotient map:

$v_1\cdots v_k\mapsto \frac{1}{k!}\sum _{\sigma \in S_k}{v}_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (k)}.$

For instance, $vw\mapsto \frac{1}{2}(v\otimes w+w\otimes v)$.

This is related to the representation theory of the symmetric group: in characteristic 0, over an algebraically closed field, the group algebra is semisimple, so every representation splits into a direct sum of irreducible representations, and if T = S ⊕V, one can identify S as either a subspace of T or as the quotient T/V.

Interpretation as polynomials

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

Given a vector space V, the polynomials on this space are S(V*), the symmetric algebra of the dual space: a polynomial on a space evaluates vectors on the space, via the pairing $S(V^{*})\times V\to K$.

For instance, given the plane with a basis {(1,0), (0,1)}, the (homogeneous) linear polynomials on K² are generated by the coordinate functionals x and y. These coordinates are covectors: given a vector, they evaluate to their coordinate, for instance:

$x(2,3)=2,\text{ and }y(2,3)=3.$

Given monomials of higher degree, these are elements of various symmetric powers, and a general polynomial is an element of the symmetric algebra. Without a choice of basis for the vector space, the same holds, but one has a polynomial algebra without choice of basis.

Conversely, the symmetric algebra of the vector space itself can be interpreted, not as polynomials on the vector space (since one cannot evaluate an element of the symmetric algebra of a vector space against a vector in that space: there is no pairing between S(V) and V), but polynomials in the vectors, such as v² - vw + uv.

Symmetric algebra of an affine space

One can analogously construct the symmetric algebra on an affine space (or its dual, which corresponds to polynomials on that affine space). The key difference is that the symmetric algebra of an affine space is not a graded algebra, but a filtered algebra: one can determine the degree of a polynomial on an affine space, but not its homogeneous parts.

For instance, given a linear polynomial on a vector space, one can determine its constant part by evaluating at 0. On an affine space, there is no distinguished point, so one cannot do this (choosing a point turns an affine space into a vector space).

Categorical properties

The symmetric algebra on a vector space is a free object in the category of commutative unital associative algebras (in the sequel, "commutative algebras").

Formally, the map that sends a vector space to its symmetric algebra is a functor from vector spaces over K to commutative algebras over K, and is a free object, meaning that it is left adjoint to the forgetful functor that sends a commutative algebra to its underlying vector space.

The unit of the adjunction is the map V → S(V) that embeds a vector space in its symmetric algebra.

Commutative algebras are a reflective subcategory of algebras; given an algebra A, one can quotient out by its commutator ideal generated by ab - ba, obtaining a commutative algebra, analogously to abelianization of a group. The construction of the symmetric algebra as a quotient of the tensor algebra is, as functors, a composition of the free algebra functor with this reflection.

Analogy with exterior algebra

The S^k are functors comparable to the exterior powers; here, though, the dimension grows with k; it is given by

$\dim (S^k(V))=(\genfrac{}{}{0ex}{}{n+k-1}{k})$

where n is the dimension of V. This binomial coefficient is the number of n-variable monomials of degree k.

Module analog

The construction of the symmetric algebra generalizes to the symmetric algebra S(M) of a module M over a commutative ring. If M is a free module over the ring R, then its symmetric algebra is isomorphic to the polynomial algebra over R whose indeterminates are a basis of M, just like the symmetric algebra of a vector space. However, that is not true if M is not free; then S(M) is more complicated.

As a universal enveloping algebra

The symmetric algebra S(V) is the universal enveloping algebra of an abelian Lie algebra, i.e. one in which the Lie bracket is identically 0.

See also

• exterior algebra, the anti-symmetric analog
• Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic form
• Clifford algebra, a quantum deformation of the exterior algebra by a quadratic form

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010