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A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K.The Clifford algebra Cl(V, Q) is the "freest" unital associative algebra generated by V subject to the condition [c] = , where the product on the left is that of the algebra, and the 1 is its multiplicative identity.
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified as rings. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions ...
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of statements within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication.
The Keystone Exam is a Pennsylvania standardized test administered to the public schools of Pennsylvania, United States. The test has been developed by the Commonwealth of Pennsylvania Department of Education. Since the 2012–2013 school year, the General Keystone Knowledge Test Literature, Biology, and Algebra I VHS Exams have been available. [1]
Ext functor. In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be ...
This ring is an R-algebra, associative and unital with identity element given by 1 A ⊗ 1 B. [3] where 1 A and 1 B are the identity elements of A and B. If A and B are commutative, then the tensor product is commutative as well. The tensor product turns the category of R-algebras into a symmetric monoidal category. [citation needed]