In
mathematics, a quadratic algebra is a
filtered algebra In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory.
A filtered algebra over the field k is an alge ...
generated by degree one elements, with defining relations of degree 2. It was pointed out by
Yuri Manin
Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical log ...
that such algebras play an important role in the theory of
quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...
s. The most important class of graded quadratic algebras is
Koszul algebra
In abstract algebra, a Koszul algebra R is a graded k-algebra over which the ground field k has a linear minimal graded free resolution, ''i.e.'', there exists an exact sequence:
:\cdots \rightarrow R(-i)^ \rightarrow \cdots \rightarrow R(-2)^ ...
s.
Definition
A graded quadratic algebra ''A'' is determined by a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of generators ''V'' = ''A''
1 and a subspace of homogeneous quadratic relations ''S'' ⊂ ''V'' ⊗ ''V'' . Thus
:
and inherits its grading from the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
''T''(''V'').
If the subspace of relations is instead allowed to also contain inhomogeneous degree 2 elements, i.e. ''S'' ⊂ ''k'' ⊕ ''V'' ⊕ (''V'' ⊗ ''V''), this construction results in a filtered quadratic algebra.
A graded quadratic algebra ''A'' as above admits a quadratic dual: the quadratic algebra generated by ''V''
* and with quadratic relations forming the orthogonal complement of ''S'' in ''V''
* ⊗ ''V''
*.
Examples
*
Tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
,
symmetric algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
and
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
of a finite-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
are graded quadratic (in fact, Koszul) algebras.
*
Universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the represent ...
of a finite-dimensional
Lie algebra is a filtered quadratic algebra.
References
*
*
Algebras
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